
    UNITED STATES of America, Plaintiff-Appellee, v. William BEAVERS, Defendant-Appellant.
    No. 13-3198.
    United States Court of Appeals, Seventh Circuit.
    Argued April 3, 2014.
    Decided June 30, 2014.
    
      Debra Riggs Bonamici, Matthew M. Getter, Office of the United States Attorney, Chicago, IL, for Plaintiff-Appellee.
    Samuel F. Adam, Chicago, IL, Victor P. Henderson, Henderson Adam, LLC, Chicago, IL, Sheldon Sorosky, Kaplan & Soro-sky, Chicago, IL, for Defendant-Appellant.
    Before POSNER, FLAUM, and ROVNER, Circuit Judges.
   FLAUM, Circuit Judge.

William Beavers, a former Chicago alderman and Cook County Commissioner, was convicted of multiple counts of tax fraud. He appeals his conviction on several grounds. Beavers challenges many of the district court’s evidentiary rulings and jury instructions, and he also contends that his right to a jury composed of a fair cross-section of the community was violated. We affirm.

I. Background

William Beavers was a Chicago aider-man from 1983 until November 2006. In December 2006, he began serving as a Cook County Commissioner. Beavers had three political campaign committees: Citizens for Beavers, Friends of William Beavers, and the 7th Ward Democratic Organization. He was the chairman of each campaign committee as well as the only authorized signor for each committee’s bank account. There is no dispute that Beavers’ federal tax returns from 2005 to 2008 exhibited various inaccuracies; the parties disagree, however, as to whether these inaccuracies were honest mistakes or deliberate misrepresentations.

The first inaccuracy was Beavers’ un-derreporting of his 2005 income. Each of Beavers’ campaign committees was required to file semi-annual disclosure reports (called “D~2s”) listing its expenditures. In its D-2 for the first half of 2005, Citizens for Beavers reported a payment of $56,149 to Beavers. This payment was legal under Illinois law. However, Beavers had given his tax preparer a letter, addressed to the IBS and signed by Beavers, stating that he had received only $43,000 in campaign funds from Citizens for Beavers. Thus, Beavers’ tax return listed $43,000 — rather than $56,149' — as “additional income.”

The second inaccuracy concerned Beavers’ undeclared use of campaign funds to increase his pension annuity. Shortly before Beavers left his position as a Chicago alderman in November 2006, the Municipal Employees Annuity and Benefit Fund of Chicago informed Beavers about his options for his aldermanic pension. To take advantage of the option that maximized his pension benefits, Beavers provided the Annuity Fund with a check for $68,763, which was drawn on the account of Citizens for Beavers. Beavers did not report the $68,763 as income on his tax return for 2006, nor did Citizens for Beavers report the expenditure on its D-2s for 2006, 2007, or 2008. Additionally, when Beavers later applied to a bank for a personal loan, his financial statements did not include the $68,763 as an outstanding loan from his campaign. From the time he took the funds in November 2006 until April 2009, Beavers did not repay any of the $68,763 to his campaign.

The third inaccuracy concerned the monthly stipends that Beavers took as a Cook County Commissioner. Cook County paid its Commissioners not only an annual salary, but also a monthly stipend of $1,200 from the Cook County Contingency Account. Beavers cashed (or deposited in his personal bank account) every monthly check from December 2006 through November 2008. Beavers informed the County (through forms that he submitted) that he would claim these $1,200 stipends as income. However, he did not report these monthly cheeks on his tax returns for 2006, 2007, or 2008.

The fourth problem was that, between 2006 and 2008, Beavers wrote himself 100 checks totaling $226,300 from his three campaign-committee accounts. At trial, the government demonstrated that Beavers often wrote these checks in order to finance his gambling trips to the Horseshoe Casino in Hammond, Indiana. Beavers cashed 93 of the 100 checks the day before, the day of, or the day after he used his “player’s card” at the Horseshoe. An IRS agent who examined these checks and the campaigns’ bank accounts testified that Beavers often repaid his campaigns in some amount, but he never repaid the full $226,300. The agent further testified that there were no loan agreements or formal documentation for any of these “advances.”

The fifth problem concerned Beavers’ efforts to obstruct the IRS. The government says that the fact that Beavers “made the 100 campaign checks payable to himself made it more difficult to determine what he did with the proceeds.” Beavers’ campaign-committees’ records said nothing of gambling or personal use. Indeed, some check stubs had no explanation of the check’s purpose, while others indicated that checks were used for campaign-related expenses even though the timing suggested that they were used to fund Beavers’ gambling. Beavers and his campaign committees did not document when he repaid the funds.

In April 2009, federal agents approached Beavers and said they wanted to interview him in connection with a grand jury investigation into his unreported conversion of campaign funds for personal use. Beavers then took several corrective actions. One week after agents contacted him, he filed amended tax returns for 2007 and 2008. He reported nearly $20,000 in additional income for each year, explaining that “[cjampaign funds deemed reportable was [sic ] inadvertently omitted from the original return.” The following month, Beavers wrote a $68,000 check — drawn on the account of one of his campaign funds, Friends for Beavers — to repay another campaign fund, Citizens for Beavers. (He later wrote several checks from his personal account in an effort to repay Friends for Beavers.) In fall 2010, the media reported that a grand jury had issued a subpoena for Cook County records about Beavers and the contingency-fund stipends. Later, in April 2011, Beavers filed a second amended tax return for 2008 in which he reported an additional $11,000 in income. This newly amended return included the explanation that “[ajdvances for expenses by the employer were not included on the W2 and were not accounted for on the original return.”

In 2012, the government charged Beavers with three counts of violating 26 U.S.C. § 7206(1), which prohibits wilfully making a material false statement on a tax return, and with one count of violating 26 U.S.C. § 7212(a), which prohibits corruptly obstructing the IRS in its administration of the tax laws. The jury convicted Beavers on all counts. Beavers was sentenced to six months’ imprisonment and was ordered to pay about $31,000 in restitution and a $10,000 fine. He appeals.

II. Discussion

A. Evidentiary rulings

Beavers raises several evidentiary challenges. He also argues that the district court’s rulings violated his constitutional right to present a meaningful defense and impermissibly burdened his right against self-incrimination. We typically review ev-identiary rulings for an abuse of discretion, but we review de novo whether an eviden-tiary ruling violated Beavers’ constitutional rights. United States v. Alayeto, 628 F.3d 917, 920-21 (7th Cir.2010).

i. The evidence of Beavers’ remedial actions

Beavers first argues that the district court erred by excluding evidence of his conduct after federal agents approached him — namely his amended tax returns and payments to reimburse his campaign committees. The government moved in limine to exclude all of this evidence. Relying on Federal Rules of Evidence 401 and 403, the government argued that Beavers’ actions were not probative of his state of mind at the time he filed the original returns, and that the jury would be confused by the • admission of evidence of remedial actions. Beavers, on the other hand, argued that such evidence was probative of his good faith and lack of intent to file fraudulent returns in the first place. He also said that the fact that he repaid (with campaign funds) the'$68,763 that he used to increase his pension shows that he considered the amount a loan, rather than income.

The district court ruled that the evidence would be admissible if Beavers could establish that each remedial action was relevant to his state of mind at the time he filed the original tax returns. Beavers ultimately elected not to testify, and he did not otherwise succeed in establishing the required evidentiary foundation. Thus, the evidence of his remedial actions was not presented at trial. Beavers argues that (1) the evidence of his remedial actions was relevant under Rule 401; (2) the district court’s conditional admission impermissibly burdened Beavers’ Fifth Amendment right against self-incrimination; and (3) the court’s rulings deprived him of his constitutional right to present a meaningful defense.

First, the district court conditioned the admission of evidence of Beavers’ corrective actions upon a showing that these actions had a connection to Beavers’ state of mind at the time he filed his incorrect returns. The logic of the ruling was that, in the absence of a foundation establishing this link, the amended tax returns (and evidence of other remedial actions) did not make it more likely that Beavers believed his original returns were accurate when he filed them.

District judges making relevancy determinations' in this type of situation should proceed on a case-by-case basis. Cf United States v. Tishberg, 854 F.2d 1070, 1073 (7th Cir.1988) (reasoning, in the sufficiency context, that the defendant’s amended return “may demonstrate a good faith effort to correct his previous mistakes,” but also noting that the return “does not negate the import of his previous action”). Nevertheless, courts have repeatedly affirmed the exclusion of evidence of remedial action taken after the taxpayer knows he is under investigation. See United States v. McClain, 934 F.2d 822, 834-35 (7th Cir.1991) (finding “no reason” to disturb the district court’s ruling that defendant’s 1985 tax return was not probative of his state of mind at the time he filed his 1984 return, given his indictment in the intervening year); United States v. Radtke, 415 F.3d 826, 840-41 (8th Cir.2005); United States v. Ross, 626 F.2d 77, 81 (9th Cir.1980); Post v. United States, 407 F.2d 319, 325 (D.C.Cir.1968); United States v. Stoehr, 196 F.2d 276, 282 (3d Cir.1952); see also United States v. Philpot, 733 F.3d 734, 748 (7th Cir.2013) (explaining, in political corruption case, that defendant’s remedial “actions after his bonuses were reported in the press shed little or no light on his state of mind two years earlier” when he received those bonuses). A common thread in many of these cases is that subsequent remedial actions may not be probative of the defendant’s prior state of mind because such actions are equally consistent with (1) promptly correcting a genuine mistake and (2) trying to cover up a purposeful lie in the hope of avoiding prosecution. Cf. Fed. R.Evid. 407 (barring evidence of subsequent remedial action to prove an admission of fault).

In any event, the district court did not exclude the evidence entirely — it simply conditioned the evidence’s admission on some kind of showing of its relevance to Beavers’ state of mind at the time he filed his original returns several years earlier. Beavers did not make this showing. In fact, his theory of defense is at odds with his argument that the remedial evidence was relevant. Beavers’ argument in defense (which he maintains on appeal) was that all of the transfers from his campaign committees were loans, not income. Thus, evidence that Beavers declared some of the campaign-fund transfers as income on his amended tax returns after he was under investigation has little bearing on whether he considered the transfers to be loans at the time he took the funds. In sum, the district court’s sensible approach to the remedial evidence was within its discretion.

Beavers next argues that the district court’s handling of this issue impermissibly burdened his Fifth Amendment right against self-incrimination, see generally Griffin v. California, 380 U.S. 609, 85 S.Ct. 1229, 14 L.Ed.2d 106 (1965), because it forced him either to testify (in order to lay the appropriate foundation) or to forgo the opportunity to get evidence before the jury. Beavers’ argument misunderstands the nature of this right, however. Criminal defendants often face difficult choices in weighing the costs and benefits of testifying. And the rules of evidence sometimes prevent defendants from getting their story — or evidence of their state of mind — before the jury in the particular manner they would prefer. Cf. U.S. ex rel. Harris v. Illinois, 457 F.2d 191, 197-98 (7th Cir.1972) (“The State presently contends, and we agree, that a defendant in a criminal trial may not introduce evidence in his defense without laying a proper foundation therefor.”). For instance, the general rule against hearsay, see Fed.R.Evid. 802, would typically prevent a defendant from calling a friend to the stand to relay an exculpatory statement the defendant said to her, in lieu of the defendant testifying himself. Thus, the rule against hearsay may burden the defendant’s right to testify in the same way Beavers argues that his right was burdened here, since it puts him to the choice of taking the stand (and exposing himself to potentially damaging cross-examination) or losing the chance to get evidence before the jury. That sort of burden is not impermissible, however; it is simply part of a larger system in which “[t]he accused does not have an unfettered right to offer testimony that is ... inadmissible under standard rules of evidence.” Taylor v. Illinois, 484 U.S. 400, 410, 108 S.Ct. 646, 98 L.Ed.2d 798 (1988). We see no meaningful difference between the hearsay situation and the burden Beavers faced here— both burdens are ordinary, well-established, and permissible. Indeed, we have previously approved the exclusion of purported state-of-mind evidence offered by a defendant where the defendant could have, but did not, supply the requisite foundation of relevance through his own testimony. See United States v. Scott, 660 F.2d 1145, 1165-67 (7th Cir.1981). Accordingly, we reject Beavers’ argument on this score.

ii. Evidence of Cook County’s alleged obligation to report the $1,200 monthly stipends on Beavers’ W-2s

Beavers next argues that the district court improperly excluded evidence about Cook County’s alleged mistakes on Beavers’ W-2s. Recall that, when Beavers was a Cook County Commissioner, the county paid him not only a salary but also a monthly stipend of $1,200. Beavers cashed or deposited in his personal bank account every monthly check from December 2006 through November 2008. He informed the county that he would claim these stipends as income, but he did not report the $1,200 monthly stipends as income on his 2006, 2007, or 2008 tax returns. The county also did not include Beavers’ monthly stipends in Beavers’ W-2s.

Beavers’ tax expert, Barry Gershinzon, was prepared to testify that in his opinion, the county should have known that Beavers was taking these stipend payments as income and should therefore have included these payments as income on Beavers’ W~ 2s. As with the evidence of Beavers’ corrective actions, the district court ruled that the evidence of the county’s supposed obligation with respect to these stipends was relevant (and thus admissible) only upon a showing that Beavers knew of that obligation at the time he completed his tax returns and relied on his W-2s as an accurate statement of his income. Beavers contends that, just as the government was allowed to introduce circumstantial evidence of his intent to lie to the IRS, so too should he have been able to offer circumstantial evidence that his failure to include these stipends on his tax returns was simply a mistake.

Beavers’ argument is unconvincing. In an ordinary case, it may well be that an individual can reasonably rely on a W-2 from his employer as an accurate statement of his income. In this case, however, the government explained at oral argument — and Beavers did not dispute — that Beavers received his 2007 and 2008 W-2s from the county before he advised the county each year that he would take the $1,200 monthly stipends as income. This timeline strongly suggests that Beavers could not have reasonably relied on a document that he knew understated his income. (And even putting the timeline aside, it is reasonable for a district court to rule that a defendant in an “intent” ease should not be able to take advantage of a fact that helps his case if the defendant did not actually know of the fact at the relevant time. See, e.g., United States v. Harris, 942 F.2d 1125, 1132 n. 6 (7th Cir.1991) (noting that testimony on the ambiguity of tax law on which defendant did not actually rely would be irrelevant if the defendant’s subjective belief were at issue).) Nonetheless, here too the court did not exclude this evidence, but instead conditioned its admission upon a showing of its relevance to Beavers’ knowledge and reliance. This ruling was well within the court’s “considerable discretion.” United States v. Marshall, 75 F.3d 1097, 1109 (7th Cir.1996) (citation omitted).

iii. Right to present a meaningful defense

Finally, Beavers contends that the district court’s evidentiary rulings deprived him of his constitutional right to present a meaningful defense. “Whether rooted directly in the Due Process Clause of the Fourteenth Amendment or in the Compulsory Process or Confrontation clauses of the Sixth Amendment, the Constitution guarantees criminal defendants a meaningful opportunity to present a complete defense.” Crane v. Kentucky, 476 U.S. 683, 690, 106 S.Ct. 2142, 90 L.Ed.2d 636 (1986) (citations and internal quotation marks omitted). But while the Constitution “prohibits the exclusion of defense evidence under rules that serve no legitimate purpose or that are disproportionate to the ends that they are asserted to promote,” Holmes v. S. Carolina, 547 U.S. 319, 326, 126 S.Ct. 1727, 164 L.Ed.2d 503 (2006), the Constitution does not require the admission of irrelevant evidence (or other types of evidence whose relevance is outweighed by other important considerations), see Crane, 476 U.S. at 689-90, 106 S.Ct. 2142; Delaware v. Van Arsdall, 475 U.S. 673, 679, 106 S.Ct. 1431, 89 L.Ed.2d 674 (1986). As established above, the district court’s repeated invocation of the relevance rule was reasonable and amply supported by precedent. We therefore find no infringement of Beavers’ right to present a meaningful defense.

B. The district court’s handling of Beavers’ tax expert

Beavers next argues that the district court erred in multiple respects in its handling of his tax expert (and sole witness), Barry Gershinzon. First, Beavers contends that the court permitted the government to conduct an overly extensive voir dire of Gershinzon, when the court should have accepted Beavers’ written disclosure as sufficient. Second, Beavers argues that the court improperly limited Gershinzon’s testimony — most importantly, his opinion as to whether the checks from Beavers’ campaign fund were loans — because the court determined that Gershinzon was unreliable. Third, Beavers maintains that the court erred by instructing the jury to partly disregard Gershinzon’s opinion. We review the district court’s actions for an abuse of discretion. See United States v. Lee, 502 F.3d 691, 698 (7th Cir.2007); United States v. Koopmans, 757 F.2d 901, 906 (7th Cir.1985).

i. Background

We will provide more detailed information below, but a brief summary of the overall proceedings may be useful. Before trial, the defense repeatedly provided insufficiently detailed summaries of Gershin-zon’s opinions as well as the bases for his views. These summaries provided lists of general topics and noted that the defense tax expert would opine that Beavers complied with the tax code, but did not disclose further required information. These insufficient disclosures led the court to allow the government to conduct a voir dire examination of Gershinzon. In the voir dire, Gershinzon testified that in his opinion, the $68,763 check from Beavers’ campaign committee to his pension fund was a loan, not income to Beavers. He also explained that in his opinion, the 100 campaign checks to Beavers were loans and advances. Gershinzon explained that, in reaching both conclusions, he relied heavily on Beavers’ statements that he (Beavers) considered the transactions to be loans. The district court ruled that Gersh-inzon could not base his expert opinions on what Beavers had told him about his (Beavers’) state of mind at the time of the charged offenses.

Gershinzon then testified before the jury, where he initially offered his opinions about whether the pension payment and 100 checks were loans. However, when Gershinzon’s testimony then veered into topics that were arguably irrelevant, the district court ordered a second voir dire examination — conducted outside the jury’s presence — to determine the relevance of those statements.

At this second voir dire, Gershinzon testified that Beavers’ W-2 from Cook County incorrectly omitted the $1,200 monthly stipend from his gross income. He also testified that he believed Beavers’ W-2 from the City of Chicago was incorrect because it did not reflect'the $68,763 campaign check to the pension fund “as a reduction 'of taxable wages.” When questioned by the government as to whether Beavers’ $68,763 contribution to the pension was “voluntary” — an issue that had tax implications — Gershinzon said the payment was not mandatory, but he did not feel comfortable opining whether it was “voluntary” on Beavers’ part.

At the end of this voir dire, the district court said that it was “deeply concerned” about Gershinzon’s reliability. The court noted that Gershinzon could not testify about the basis for his conclusions without reliance on Beavers’ statements to him about Beavers’ subjective intent. The court also stated that Gershinzon could not “follow the rules” and was not “careful with his assumptions.” As an example, the court cited Gershinzon’s testimony that many of his clients do not understand their W-2s, but said that Gershinzon did not consider the possibility that a former alderman and Cook County Commissioner could “get a detailed explanation from high ranking City and County officials as to what was in his W-2 and how it was arrived at.”

The court ultimately struck Gershinzon’s conclusions about whether the transactions were loans, but it still allowed the jury to consider Gershinzon’s testimony with respect to the objective characteristics Gershinzon would look for in determining whether a transfer was a loan, an advance, or income.

ii. The first voir dire of Gershinzon

Beavers argues that the court should have accepted his written disclosures as sufficient and that the court thus erred in permitting the government to conduct an extensive voir dire of Gershinzon before trial. Federal Rule of Criminal Procedure 16 required Beavers to provide the government with a written summary of any expert testimony that he intended to use as evidence at trial. Fed. R.Crim. Pro. 16(b)(1)(C). This summary must “describe the witness’s opinions, the bases and reasons for those opinions, and the witness’s qualifications.” Id.

Before trial, the defense repeatedly gave the government insufficiently detailed summaries. These summaries provided a general list of examination topics (e.g., “general accounting principles,” “general tax reporting and bookkeeping”) and one letter noted — in one sentence — that Gersh-inzon would testify that in his opinion Beavers complied with the tax code. The summaries did not provide any more specific information about Gershinzon’s opinions or the bases therefore. We agree with the district court that the summaries were “sketchy on details” and that the voir dire was an appropriate remedy. In fact, Beavers arguably chose that course. (During a break in the voir dire, defense counsel said, “Your Honor gave us a choice of either submitting documents or voir dire. I think we chose — we chose the voir dire....”) And in any event, Beavers must — but cannot — show prejudice from the court’s decision to allow the voir dire, see United States v. White, 582 F.3d 787, 804 (7th Cir.2009), because he was not entitled to surprise the government with ill-defined expert testimony, see United States v. Hoffecker, 530 F.3d 137, 185 (3d Cir.2008) (noting that Rule 16 seeks “to minimize surprise that often results from unexpected expert testimony, reduce the need for continuances, and to provide the opponent with a fair opportunity to test the merit of the expert’s testimony through focused cross-examination.” (quoting Rule 16 Advisory Committee Notes (1993 Amendment))). We therefore reject Beavers’ argument that the court abused its discretion on this score.

iii. Limits on Gershinzon’s testimony

Beavers next disputes the district court’s various limitations on Gershinzon’s testimony. ‘We review a district court’s decision to admit or exclude expert testimony for abuse of discretion,” keeping in mind that screening evidence “is a function squarely within the purview of the trial judge.” Lapsley v. Xtek, Inc., 689 F.3d 802, 809 (7th Cir.2012). We can conceptualize some of the limitations as state-of-mind limitations, others as relevance limitations, and still others as reliability limitations.

We begin with state-of-mind limitations. First, the district court ruled that Gershinzon could not base his expert opinions on what Beavers had told him about his (Beavers’) state of mind at the time of the charged offenses. This limitation was proper. Otherwise, Beavers could have gotten highly selective and favorable statements of his before the jury without having to face cross-examination. Later, the district court also ruled that Gershinzon could not testify as to whether he (Gershinzon) believed the 100 checks were loans. The court reasoned that such testimony would violate Rule 704(b) of the Federal Rules of Evidence, which prohibits expert witnesses in a criminal case from opining as to whether the defendant had the mental state necessary for the charged crime. As the judge explained, “[t]he relevant state of mind is whether defendant intended to treat withdrawals from his campaign coffers as income or as loans. The expert cannot state an opinion on this issue.” For Gershinzon to have addressed this issue would have been the equivalent of opining on whether Beavers had the “willfulness” necessary for a tax offense. See United States v. Windfelder, 790 F.2d 576, 582 (7th Cir.1986). We accordingly find no abuse of discretion in this limitation.

Next, the relevance limitations prohibited Gershinzon from testifying about issues that were not relevant absent testimony or other evidence connecting them to Beavers’ state of mind at the time of the charged offenses. For example, the court barred Gershinzon from offering his opinion that Cook County erred by omitting the $1,200 monthly stipend payments to Beavers. As explained above, this testimony was irrelevant absent evidence that Beavers knew of and relied on the county’s alleged obligation. Similarly, the defense sought to ask Gershinzon about a form that Beavers received from the pension fund in early 2008, as part of the defense’s attempt to show that Beavers’ $68,763 pension payment was not income. The court prohibited Gershinzon from testifying about this form because Beavers received it in 2008, so it was irrelevant to Beavers’ state of mind in 2007 when he filed his 2006 tax return. The court’s relevance limitations were therefore appropriate. (And where Gershinzon’s testimony was relevant, the court permitted it. For instance, Gershinzon testified about accounting principles; about the purpose of certain tax forms; about how tax professionals disagree on the proper tax treatment of certain transactions; about the differences between income, loans, and advances; and about which records he would consider in determining whether a transfer was income, a loan, or an advance.)

We turn now to the court’s reliability limitations. Recall, the court’s determination as to Gershinzon’s reliability ultimately resulted in the court’s finding Gershinzon unreliable and instructing the jury to partly disregard Gershinzon’s testimony — including, importantly, his opinions as to whether the $68,763 pension payment and the 100 checks were loans or income.

The court found Gershinzon unreliable for multiple reasons. Crucially, the court found that Gershinzon “[wa]s not careful with his assumptions,” and those “assumptions have overtaken at least the minimum neutrality an expert is supposed to have.” As noted above, one assumption was that, because many of Gershinzon’s clients do not understand their W-2s, it apparently followed that a high-ranking public official (with access to advisors) was in the same position. The court also criticized Gersh-inzon’s assumptions because Gershinzon relied on his conversations with Beavers in forming assumptions and ultimately conclusions about the proper tax treatment of the transactions at issue in this ease. For instance, in concluding that the $68,763 pension payment was a loan, Gershinzon relied in substantial part on Beavers’ statement to Gershinzon that Beavers considered it a loan. Similarly, in opining that the 100 campaign checks to Beavers were loans and advances, Gershinzon also relied on Beavers’ statements to him. Even though the court had instructed Gershinzon not to rely on Beaversi statements, Gershinzon explicitly referenced those statements, indicating to the court that Gershinzon could not get Beavers’ statements “out of his [own] mind.... He can’t follow the rules.” Gershinzon was also unable to point to portions of the tax code to support his assertions about the tax treatment of various payments, including the pension-fund payment. Thus, stripped of Beavers’ statements, Gershinzon’s most important opinions lacked meaningful support.

Gershinzon also was unable to answer basic questions about the topic of his testimony. Specifically, during the second voir dire, the government asked Gershinzon about the basis for his opinion that the City of Chicago should have deducted Beavers’ lump-sum pension payment from his gross income for 2006. Gershinzon acknowledged that Beavers’ pension payment was not mandatory, but was unable to say whether it was “voluntary,” because he could not testify to Beavers’ state of mind. The prosecutor clarified that this was not a state-of-mind question; rather, a letter gave Beavers three options, and Beavers selected the option that maximized his pension benefits. Nonetheless, Gershinzon would not say that Beavers’ choice was “voluntary.” After considerable back-and-forth between the prosecutor and Gershinzon, the court eventually said, “[a] step which is not compelled is a step which is voluntary.”

We “give the district court wide latitude in performing its gatekeeping function and determining both how to measure the reliability of expert testimony and whether the testimony itself is reliable.” Bielskis v. Louisville Ladder, Inc., 663 F.3d 887, 894 (7th Cir.2011). In light of the foregoing analysis — which indicates that Beavers’ expert relied on irrelevant considerations, made questionable assumptions, and may have lacked expertise about certain germane subjects (including relevant portions of the tax code) — we find that the district court properly exercised its discretion. Moreover, the court still allowed Gershin-zon to testify as to the objective characteristics he would look for in determining whether a particular transfer was income, a loan, or an advance. As a result, Gersh-inzon’s analytical methodology was presented to the jury, even if his particular conclusions were not.

iv. Jury instruction about Gershinzon’s conclusions

As noted, after Gershinzon testified, the district court instructed the jury to disregard his opinions as to whether the relevant transactions were loans, advances, or income. However, the court also instructed the jury that it could consider Gershinzon’s testimony about his methodology in determining whether a particular transfer is a loan versus income. Beavers argues that the court’s wording was inappropriate because it included a reference to the government expert’s testimony; according to Beavers, this instruction “directed] the jury to disregard the defense expert but to consider the government’s expert.” Contrary to Beavers’ argument, the instruction expressly told the jurors that they could “consider Mr. Gershinzon’s testimony, as well as the testimony of the government’s [expert] witness, David Weiner, regarding facts and circumstances relevant in deciding whether a particular check is a loan, advance, or income.” Beavers’ argument is therefore unpersuasive, and the court’s instruction was well within its discretion.

C. Jury instruction about the definition of a loan

Next, Beavers contends that the district court incorrectly instructed the jury on the definition of a “loan.” We review the legal accuracy of a jury instruction de novo, but we evaluate the particular phrasing for abuse of discretion. United States v. Dickerson, 705 F.3d 683, 688 (7th Cir.2013).

Beavers wanted the following instruction:

When a taxpayer receives a loan, he incurs an obligation to repay that loan at some future date. Because of this obligation, the loan proceeds do not qualify as income to the taxpayer. When he fulfills the obligation, the repayment of the loan likewise has no effect on his tax liability.

Beavers’ proposed instruction quotes the Supreme Court’s language in a tax case, C.I.R. v. Tufts, 461 U.S. 300, 307, 103 S.Ct. 1826, 75 L.Ed.2d 863 (1983). The district court instead gave the following instruction, which the government proposed:

When a taxpayer receives a loan, he incurs an obligation to repay that loan. Because of that obligation to repay, loan proceeds do not constitute income. The transfer of money from one party to another constitutes a loan only if, at the time of the transfer, the parties to the transaction intend that the person who receives the money actually will be obligated to repay it.
In determining whether the defendant has received particular funds as a loan or as income, you should consider all of the facts and circumstances surrounding the defendant’s receipt of the funds.

Both instructions are correct statements of the law. Both convey that loan proceeds are not income because the taxpayer has incurred a genuine obligation to repay the loan.

Nonetheless, Beavers argues that the district court’s instruction was erroneous for two reasons. First, he says that the court was wrong to include the word “actually” in instructing the jury that a transfer of money is a loan only if “the person who receives the money actually will be obligated to repay it.” (emphasis added). Beavers states that the word “actually” is “superfluous” because an obligation to repay a loan means an actual obligation to repay that loan. Appellant Br., 28. But superfluity is not error. By pointing out that an obligation is an actual obligation, Beavers concedes that the court’s statement of the law is correct. The court’s inclusion of an additional (perhaps unnecessary) term for emphasis does not transform a correct statement of the law into an incorrect one, and it falls well within the court’s discretion. Relatedly, Beavers argues that the court’s defining a loan as an “actual obligation” implied that the parties’ subjective intent to repay is not enough, and that some tangible obligation (like a promissory note) is required. But the instruction simply does not say that. Rather, it conveys that the recipient must actually intend to repay, which is legally accurate and consistent with case law. See, e.g., Tufts, 461 U.S. at 307, 103 S.Ct. 1826 (using the phrase “true loan”); Crowley v. C.I.R., 962 F.2d 1077, 1082 (1st Cir.1992) (using “bona fide loan”); Frierdich v. C.I.R., 925 F.2d 180, 185-86 (7th Cir.1991) (using both “true loan” and “bona fide loan”); Moore v. United States, 412 F.2d 974, 978 (5th Cir.1969) (using “bona fide loan”).

Second, Beavers argues that the district court erred by instructing the jury that something is a loan only if the parties intend it as such at the time of the transfer. Beavers raised this objection after the instruction was given, so we review his claim only for plain error. See United States v. Natale, 719 F.3d 719, 729 (7th Cir.2013). We will reverse only if there was an obvious error that affected Beavers’ substantial rights, and that error seriously affects the fairness or integrity of judicial proceedings. United States v. Marcus, 560 U.S. 258, 262, 130 S.Ct. 2159, 176 L.Ed.2d 1012 (2010). In this case, it is not “obvious” that any error has occurred, especially because the instruction was probably correct. See, e.g., Geftman v. C.I.R., 154 F.3d 61, 68 (3d Cir.1998) (“For ‘disbursements to constitute true loans there must have been, at the time the funds were transferred, an unconditional obligation on the part of the transferee to repay the money, and an unconditional intention on the part of the transferor to secure repayment.’ ”) (quoting Haag v. C.I.R., 88 T.C. 604, 615-16 (T.C.1987), aff'd, 855 F.2d 855 (8th Cir.1988) (table)); Todd v. C.I.R., 101 T.C.M. (CCH) 1603 (T.C.2011) (“[T]he Court gives the promissory note little weight. This factor indicates the parties did not intend to establish a debtor-creditor relationship at the time the funds were advanced.”), aff'd, 486 Fed.Appx. 423 (5th Cir.2012). We therefore reject all of Beavers’ challenges to the jury instruction about the definition of a loan.

D. Fair cross-section claim

Finally, Beavers alleges that his right to a jury made up of a fair cross-section of the community was violated because none of the fifty prospective jurors were African-American males. Beavers raises both a statutory claim, based on the Jury Selection and Service Act of 1968, 28 U.S.C. §§ 1861, et seq., and a constitutional challenge based on the Sixth Amendment.

i. Background

The panel of potential jurors in this case was drawn pursuant to the Plan for Random Selection of Jurors of the United States District Court for the Northern District of Illinois in effect at the time of trial. The panel was comprised of approximately fifty potential jurors, four of whom were African-American females. Three of the four African-American prospective jurors were seated, two as jurors, and one as an alternate. The fourth was excused for cause because she worked as a tax preparer for H & R Block. After jury selection commenced, Beavers objected to the fact that no African-American men were included, and argued that this violated Bat-son v. Kentucky, 476 U.S. 79, 106 S.Ct. 1712, 90 L.Ed.2d 69 (1986). Beavers requested that the panel be dismissed and replaced with a new one or, alternatively, that the panel be supplemented with additional prospective jurors. The district court declined both requests. It noted that the relevant question was how the panel was assembled by the jury office, and said that if Beavers wished to make an argument about that office’s methodology, he could investigate the issue.

Beavers made no further requests for relief at the time; the jury was empaneled and the trial commenced. After his conviction, Beavers raised the fair cross-section issue again in his post-trial motions, arguing that the absence of African-American males in the panel was the product of “systematic exclusion.” The district court denied Beavers’ motion on the grounds that (1) it was untimely; (2) no claim or evidence of error in the administration of the plan was presented; (3) African-American males are not a recognized distinctive group; and (4) there was no evidence that the composition of the venire was brought about by any illegality or unfairness.

ii. Statutory claim

The Jury Selection and Service Act provides, “No citizen shall be excluded from service as a grand or petit juror in the district courts of the United States on account of race, color, religion, sex, national origin, or economic status.” 28 U.S.C. § 1862. A statutory challenge must be made by motion “before the voir dire examination begins, or within seven days after the defendant discovered or could have discovered, by the exercise of diligence, the grounds therefor, whichever is earlier.” 28 U.S.C. § 1867(a). The motion must also contain “a sworn statement of facts which, if true, would constitute a substantial failure to comply with the provisions of this title.... ” 28 U.S.C. § 1867(d). This is “the exclusive means” by which a violation of the Act may be raised. 28 U.S.C. § 1867(e).

It is undisputed that Beavers did not file a motion before the voir dire examination began, which is the relevant moment in this case. Instead, Beavers orally objected during the voir dire examination. He also did not provide the required “sworn statement of facts.” 28 U.S.C. § 1867(d). Thus, he waived any statutory challenge. See United States v. Phillips, 239 F.3d 829, 841 (7th Cir.2001) (“Defendants’ failure to make a motion in a timely manner and failure to provide evidence, other than oral observations as to the lack of statistical proportionality, precluded a statutory challenge.”) (citations omitted).

iii. Constitutional claim

Beavers next raises a constitutional challenge to the composition of the jury pool, which we review de novo. United States v. Neighbors, 590 F.3d 485, 491 (7th Cir.2009). “The Sixth Amendment secures to criminal defendants the right to be tried by an impartial jury drawn from sources reflecting a fair cross section of the community.” Berghuis v. Smith, 559 U.S. 314, 130 S.Ct. 1382, 1387, 176 L.Ed.2d 249 (2010). To make a prima facie showing that the fair cross-section requirement has been violated, a defendant must show that: (1) the group allegedly excluded is a distinctive group in the community, (2) the representation of this group in venires from which juries are selected is not fair and reasonable in relation to the number of such persons in the community, and (3) this underrepresentation is due to systematic exclusion of the group in the jury selection process. Duren v. Missouri, 439 U.S. 357, 364, 99 S.Ct. 664, 58 L.Ed.2d 579 (1979); Neighbors, 590 F.3d at 491.

Beavers did not develop an argument in his opening brief as to why African-American men constitute a distinctive group in the community. Women are a distinctive group, see, e.g., Duren, 439 U.S. at 364, 99 S.Ct. 664, as are African Americans, see, e.g., Neighbors, 590 F.3d at 491. But the parties have cited no cases, and we have found none, addressing whether African-American men constitute a distinctive group under Duren. We have clearly set out the standards by which we decide whether a group is “distinctive” for Duren cross-section claims, see United States v. Raszkiewicz, 169 F.3d 459, 463 (7th Cir.1999), but Beavers did not conduct that analysis or even cite that case until his reply brief.

Beavers waived this issue because he did not address this question of first impression in detail in his opening brief. See Mahaffey v. Ramos, 588 F.3d 1142, 1146 (7th Cir.2009) (“Perfunctory, undeveloped arguments without discussion or citation to pertinent legal authority are waived.”). His approach prevented the opposing party from having the opportunity in its brief to respond and fully air the arguments on the other side. We express no view on the merits of Beavers’ waived argument.

We also find that if Beavers sought to make a separate argument under the Equal Protection Clause, he waived this argument as well. His brief on appeal seems to argue that the district court committed an equal protection violation in handling jury selection as it did, because around the time of Beavers’ trial, another judge in the same district ordered a new panel when the original jury pool contained only one African-American male. However, Beavers did not develop his legal theory and cited no cases in support of it. Again, this constitutes waiver. E.g., United States v. Holm, 326 F.3d 872, 877 (7th Cir.2003).

III. Conclusion

We Affirm William Beavers’ conviction.  