BARRON, Circuit Judge.
On October 28, 2014, Robel Phillipos was convicted under 18 U.S.C. § 1001(a)(2) on two counts of making false statements to federal authorities in the weeks following the Boston Marathon bombing. The bombing occurred on April 15, 2013. The statements related to Phillipos’s possible participation, three days later, in the removal and disposal of a backpack thought to contain evidence related to the attack from the college dormitory room of Dzhokhar Tsarnaev, one of the bombing suspects and a'Mend of the defendant’s at college. Phil-lipos was sentenced to three years’ imprisonment and three years’ supervised release. He now challenges his convictions on a number of grounds that we will address, along with the facts relevant to each, in turn. Because we find that none of these challenges has merit, we affirm.
I.
We start with Phillipos’s challenges to the admission into evidence of a signed confession, in which Phillipos admitted to making the false statements that are at issue during two informal interviews with federal agents in the two weeks following the bombing. Phillipos signed that confession at the conclusion of an interview with an agent of the Federal Bureau of Investigation (“FBI”) on April 26, 2013. Phillipos contends that the District Court erred in both (1) refusing to conduct a preliminary
hearing on the admissibility of the confession unless Phillipos would agree to submit to cross-examination at that hearing on the contents of an affidavit that he submitted regarding the circumstances of the confession, and (2) failing to make a determination prior to the introduction of the confession into evidence as to whether Phillipos made the confession voluntarily. Neither challenge warrants reversal of the convictions.
A.
We begin with the first of Phillipos’s challenges to the admissibility of the confession, which concerns the denial of his request for an evidentiary hearing on the voluntariness of his confession because he refused to submit to cross-examination. Phillipos sought the hearing in connection with his motion to suppress the confession pursuant to the Fifth Amendment’s Due Process Clause.
Phillipos acknowledged that he had been informed of his rights, as required by Miranda v. Arizona, 384 U.S. 436, 86 S.Ct. 1602, 16 L.Ed.2d 694 (1966), prior to making the confession. But he contended that the circumstances under which he made it were coercive. In support of this motion, Phillipos relied solely on his own affidavit recounting his version of those circumstances.
Specifically, Phillipos’s affidavit alleged that the interviewing FBI agent, Michael Delapena, “interrogated [Phillipos] for several hours in a small room.” Phillipos also alleged that, during that time, Delapena administered a polygraph test, and told Phillipos that Phillipos could only answer questions with “yes or no.” As a result, Phillipos alleged, he answered “no” to certain questions because he could not give the truthful answer, which would have been that he did not remember. Phillipos also alleged that Delapena did not offer Phillipos food; that Delapena “got close to [Phillipos’s] face” and cursed at him; and that Delapena locked the door and told Phillipos, “[T]ell me everything that happened. There are wolves outside the door, you don’t want me to unlock the door.” In addition, Phillipos alleged that, at the end of the interrogation, Delapena presented him with a typed confession to sign, which Phillipos signed because “[he] felt that [he] had no choice but to sign it if [he] were to leave without being arrested.”
Phillipos argued that the affidavit he submitted sufficed to show that there were facts in dispute regarding the confession’s voluntary nature, because the account in the affidavit conflicted in key respects with the account the government set forth in its opposition to his motion to suppress the confession that it had filed. In its filing, the government did not dispute that Delapena questioned Phillipos for several hours in a small room.
The government did, however, dispute other aspects of Phillipos’s account. Specifically, the government stated that, prior to administering the polygraph, Delapena explained to Phillipos that the polygraph questions had to be answered with a “yes or no,” and that if Phillipos did not remember something, Delapena would rephrase the question so that Phillipos could truthfully answer with a “yes” or “no.” The government’s account also conflicted with Phillipos’s in the following respects: De-lapena did offer Phillipos food; Delapena “at no time ... raise[d] his voice, curse[d], or otherwise treatfed] the defendant discourteously”; Delapena locked the door to the interview room only after Phillipos expressed concern that agents outside were angry with him, and, in doing so, Delapena told Phillipos, “Don’t worry about them. They’re outside; I’m here with you. I don’t judge you.” Finally, the government stated in its filing that Phillipos sat with Delape-
na as Delapena typed up the confession, that Delapena “conferred constantly” with Phillipos to ensure that Delapena was accurately setting forth Phillipos’s account, and that, at several points, Phillipos asked Delapena to make specific changes to the account, which Delapena did.
A defendant is entitled to an evi-dentiary hearing as to the voluntariness of his confession only if the defendant “makes a sufficient threshold showing that material facts are in doubt or dispute, and that such facts cannot reliably be resolved on a paper record.” United States v. Staula, 80 F.3d 596, 603 (1st Cir. 1996). Applying that requirement to Phillipos’s request for the hearing, the District Court acknowledged that the allegations in Phillipos’s affidavit, if credited, would suffice to establish a factual dispute that would warrant a hearing. But the District Court went on to explain that the affidavit could suffice to establish that factual dispute only if Philli-pos agreed to be cross-examined about the affidavit’s contents at the hearing. Otherwise, the District Court ruled, the affidavit “cannot be tested” and would be “illusory.” And, as a result, there would be no basis for finding that Phillipos had established the requisite factual dispute.
We review a preserved challenge to a denial of a request for a preliminary hearing for abuse of discretion. United States v. Jiménez, 419 F.3d 34, 42 (1st Cir. 2005). “Abuse of discretion occurs ‘when a relevant factor deserving of significant weight is overlooked, or when an improper factor is accorded significant weight, or when the court considers the appropriate mix of factors, but commits a palpable error of judgment in calibrating the decisional scales.’” Id. at 43 (quoting United States v. Gilbert, 229 F.3d 15, 21 (1st Cir. 2000)). “Within this framework, an error of law is always tantamount to an abuse of discretion.” Torres-Rivera v. O’Neill-Cancel, 524 F.3d 331, 336 (1st Cir. 2008).
Phillipos contends that the District Court abused its discretion by making his willingness to submit to cross-examination a condition of holding the hearing. Philli-pos contends that the condition effectively — and impermissibly — “[f]orc[ed] him to choose between” asserting his constitutional right against self-incrimination and asserting his right to an evidentiary hearing regarding the voluntariness of his confession. But, we do not agree.
Phillipos points to nothing in the record besides his affidavit that could have provided a basis for finding that he had met his burden of demonstrating a factual dispute that would warrant an evidentiary hearing at the time that the District Court imposed the cross-examination condition. In fact, when the District Court asked one of Phillipos’s attorneys, prior to denying the motion for the hearing, to “[pjoint [the court] to” evidence in the record besides Phillipos’s own affidavit that would support Phillipos’s contention that his confession was not voluntary, the attorney failed to do so.
As a result, the District Court simply determined that Phillipos failed to establish a sufficient threshold basis for finding that there was a factual dispute over the voluntariness of his confession because he had proffered only an untested — and thus illusory — affidavit. In doing so, the District Court followed a course that we approved, albeit in a different context, in United States v. Baskin, 424 F.3d 1 (1st Cir. 2005), which is a case that Phillipos fails to address.
There, a defendant relied solely on factual allegations set forth in his own affidavit in moving on Fourth Amendment grounds to suppress the fruits of a search. Id. at 3. The defendant refused, however, to submit to cross-examination that would allow the government to “test the truth” of the factual allegations set forth in the affidavit; the defendant instead invoked his Fifth Amendment privilege against self-incrimination. Id. In consequence, the District Court refused to credit the affidavit and then denied the motion to suppress. Id. We held that the district court did not abuse its discretion in so ruling, because the district court was entitled to conclude that the defendant’s own untested affidavit was not in and of itself sufficient to “establish any ground for asserting a Fourth Amendment right.” Id.
To be sure, Baskin involved a district court’s denial, after an evidentiary hearing, of a defendant’s motion to suppress evidence allegedly acquired in violation of the defendant’s Fourth Amendment right against an unreasonable search. Id. By contrast, here we confront a defendant’s motion for an evidentiary hearing to determine whether, under the Fifth Amendment, a confession was voluntarily made. But, like the defendant in Baskin, Phillipos sought to establish the requisite factual predicate for his motion solely on the basis of his own affidavit, which he would not allow the government to test through cross-examination. Nor did Phillipos identify any additional evidence that he would be able to present at the hearing, other than his speculative assertions that he might have established support for his affidavit’s account through his own cross-examination of the government agents who conducted the interviews that led to the confession.
In light of Baskin, we see no basis for concluding that the District Court abused its discretion in finding that the affidavit, on its own, failed to establish the sufficient threshold showing of a factual dispute that Phillipos was required to make. As we have said before, “[t]he district court has considerable discretion in determining the need for, and the utility of, evidentiary hearings, and we will reverse the court’s denial of an evidentiary hearing in respect to a motion in a criminal case only for manifest abuse of that discretion.” Staula, 80 F.3d at 603. Accordingly, this first challenge regarding the use of the confession at trial fails.
B.
Phillipos next contends that the District Court erred by failing to make a finding as to the voluntariness of the confession before admitting it into evidence. Phillipos does not contend that the District Court was required to find that Phillipos’s confession was involuntary, as he concedes that the record as it then stood supported the voluntariness finding made. Phillipos contends only that the timing of the District Court’s voluntariness determination-prejudiced Phillipos’s ability to develop a complete record in support of his conten
tion that the confession was not voluntarily-given.
Phillipos failed to raise this objection below, and so our review is only for plain error. United States v. Piano, 507 U.S. 725, 732, 113 S.Ct. 1770, 123 L.Ed.2d 508 (1993). “Review for plain error entails four showings: (1) that an error occurred (2) which was clear or obvious and which not only (3) affected the defendant’s substantial rights, but also (4) seriously impaired the fairness, integrity, or public reputation of judicial proceedings.” United States v. Duarte, 246 F.3d 56, 60 (1st Cir. 2001).
Although the District Court made a preliminary finding pre-trial that the confession was voluntary, the government does not appear to dispute that the District Court erred in failing to make a conclusive voluntariness determination “[bjefore [the] confession [was] received in evidence.” 18 U.S.C. § 3501(a). See also Sims v. State of Georgia, 385 U.S. 538, 543-44, 87 S.Ct. 639, 17 L.Ed.2d 593 (1967) (“[A] jury is not to hear a confession unless and until the trial judge has determined that it was freely and voluntarily given.”); Crane v. Kentucky, 476 U.S. 683, 687-88, 106 S.Ct. 2142, 90 L.Ed.2d 636 (1986) (“To assure that the fruits of such techniques are never used to secure a conviction, due process also requires ‘that a jury [not] hear a confession unless and until the trial judge ... has determined that it was freely and voluntarily given.’ ” (alteration in original) (quoting Sims, 385 U.S. at 543-44, 87 S.Ct. 639)); United States v. Feliz, 794 F.3d 123, 130 (1st Cir. 2015) (“Although the judge need not make formal findings of fact or write an opinion, his conclusion that the confession is voluntary must appear from the record with unmistakable clarity.” (quoting Sims, 385 U.S. at 544, 87 S.Ct. 639)). But, as Phillipos identifies no way in which the timing of the District Court’s voluntariness determination affected his substantial rights, we agree with the government that there is no plain error. See Duarte, 246 F.3d at 60.
Phillipos argues otherwise on the ground that the late timing of the volun-tariness determination interfered with his ability to make his case for suppressing the confession pre-trial, as he contends that it would have been easier for him to make his case for suppression at that time. But Phillipos did have the chance to make that pre-trial case. He simply failed at that time to establish any factual dispute with the government’s account. Thus, the fact that the District Court made only a preliminary voluntariness determination prior to the start of the trial did not prejudice Phillipos. Accordingly, this version of his challenge concerning the use of the confession at trial also fails.
II.
Phillipos next contends that the District Court erred by excluding testimony from Phillipos’s proposed expert on false confessions, Dr. Richard Leo, without first conducting a hearing under Daubert v. Merrell Dow Pharmaceuticals, Inc., 509 U.S. 579, 113 S.Ct. 2786, 125 L.Ed.2d 469 (1993), as to whether Dr. Leo’s testimony qualified as expert testimony. We do not agree.
Daubert establishes that before admitting expert testimony, the trial court must fulfill its “gatekeeping role” by making an independent determination that the expert’s proffered scientific knowledge is both reliable and relevant. 509 U.S. at 597, 113 S.Ct. 2786. Daubert.also explains that the reliability determination “entails a preliminary assessment of whether the reasoning or methodology underlying the testimony is scientifically valid and of whether that reasoning or methodology
properly can be applied to the facts in issue.” Id. at 592-93, 113 S.Ct. 2786.
Phillipos contends that such an assessment may be made only after a hearing. There is, however, no such requirement. Phillipos identifies no precedent that supports his view. And we have made clear that “[tjhere is no particular procedure that the trial court is required to follow in executing its gatekeeping function under Daubert.” Smith v. Jenkins, 732 F.3d 51, 64 (1st Cir. 2013) (internal quotation marks and citation omitted); see also Kumho Tire Co. v. Carmichael, 526 U.S. 137, 152, 119 S.Ct. 1167, 143 L.Ed.2d 238 (1999) (affirming district court’s Daubert inquiry where the court did not hold a hearing because “[tjhe trial court must have the same kind of latitude in deciding how to test an expert’s reliability ... as it enjoys when it decides whether or not that expert’s relevant testimony is reliable” (emphasis in original)); United States v. Nichols, 169 F.3d 1255, 1262-64 (10th Cir. 1999) (a district court has discretion not to hold a pretrial evidentiary reliability hearing in carrying out its gatekeeping function); Hopkins v. Dow Corning Corp., 33 F.3d 1116, 1124 (9th Cir. 1994) (“The district court is not required to hold a ... hearing, but rather must merely make a determination as to the proposed expert’s qualifications.”).
The only cases that Phillipos relies on in arguing otherwise are not on point. Each held only that the district court erred in excluding testimony under Daubert because the district court had not considered the Daubert factors at all. See United States v. Hall, 93 F.3d 1337 (7th Cir. 1996); United States v. Belyea, 159 Fed.Appx. 525 (4th Cir. 2005) (per curiam) (unpublished).
The District Court made no such mistake here. It provided both Phillipos and the government multiple opportunities to explain why Dr. Leo’s testimony should or should not be admitted under Daubert. Moreover, when the District Court ulti.mately decided to exclude Dr. Leo’s testimony, the District Court explained that it had given a fair amount of thought to the issue and provided a detailed explanation of why it was excluding the testimony under the Daubert factors. It is thus clear to us that the District Court performed the gatekeeping role that Daubert requires.
Phillipos does appear to contend, separately, that the District Court erred in performing that gatekeeping role by ruling that Dr. Leo’s testimony was insufficiently reliable and thus must be excluded. Because Phillipos preserved this challenge, our review is for abuse of discretion. Smith, 732 F.3d at 64. We find none.
Under Daubert, “[t]o determine whether an expert’s testimony is sufficiently reliable, the trial court considers whether ‘the testimony is based on sufficient facts or data’; whether ‘the testimony is the product of reliable principles and methods’; and whether ‘the expert has reliably applied the principles and methods to the facts of the case.’ ” Id. at 64 (quoting Fed. R. Evid. 702(b-d)). The trial court may also consider other factors, “including but not limited to the verifiability of the expert’s theory or technique, the error rate inherent therein, whether the theory or technique has been published and/or subjected to peer review, and its level of acceptance within the scientific community.” Samaan v. St. Joseph Hosp., 670 F.3d 21, 31-32 (1st Cir. 2012) (internal quotation marks and citation omitted).
In this case, the District Court considered competing analyses of the reliability of Dr. Leo’s studies and found that there was “no indication that there is a body of reliable material that constitutes understanding in this area,” and that “it would
introduce the jury ... to 'a kind of faux science to present Dr. Leo’s testimony.” Whether or not we would reach the same conclusion, the record certainly shows that the one that the District Court reached is reasonable. Cf. People v. Kowalski, 492 Mich. 106, 821 N.W.2d 14, 31-32 (2012) (no abuse of discretion where trial court concluded that Dr. Leo’s testimony relied on “unreliable” sources and used “unreliable methodology” that “resulted in conclusions consistent with Leo’s own preconceived beliefs rather than testable results consistent with an objective, scientific process”); State v. Rafay, 168 Wash.App. 734, 285 P.3d 83, 112 (2012) (no abuse of discretion where “Leo was unable to- testify about any meaningful correlation between specific interrogation methods and false confessions or provide any method for the trier of fact to analyze the effect of the general concepts on the reliability of the defendants’ confessions”); Vent v. State, 67 P.3d 661, 667-670 (Alaska Ct. App. 2003) (no abuse of discretion where trial court concluded that “there was no way to quantify or test Dr. Leo’s conclusions that certain techniques might lead to false confessions”).
III.
Finally, and most significantly, Phillipos appeals the District Court’s denial of his motion for acquittal. The jury found Philli-pos guilty of two counts of violating 18 U.S.C. § 1001(a)(2), which applies to those who “make[] any materially false, fictitious, or fraudulent statement or representation” to federal authorities.
Specifically, the jury found Phillipos guilty, under Count One of the indictment, of making the following false statements to a federal agent during an interview on April 20, 2013: (1) Phillipos did not remember going to Tsarnaev’s dormitory room on the evening of April 18; and (2) Phillipos went to the room with Dias Kadyrbayev and Azamat Tazhayakov, two friends of his and Tsarnaev’s from college, on the evening of April 18, but no one entered the room.
The jury also found Phillipos guilty, under Count Two of the indictment, of making the following three false statements during an April 25, 2013 interview with a federal agent: (1) Phillipos had only entered Tsarnaev’s dormitory room on one occasion on April 18, which was sometime in the afternoon when he spoke to Tsar-naev for approximately ten minutes; (2) neither Phillipos, Kadyrbayev, or Tazhaya-kov took a backpack from Tsarnaev’s room on the evening of April 18; and (3) Philli-pos was nbt aware of Kadyrbayev or Ta-zhayakov removing anything from Tsar-naev’s room on the evening of April 18.
On appeal, Phillipos does not contest the falsity of these statements. Instead, he makes three separate arguments as to why the denial of his motion for acquittal must be reversed. First, he contends that there was insufficient evidence to support a finding that these statements, even though false, were ones to which section 1001 applies. Second, he contends that there was insufficient evidence to support a finding that he made the statements “knowingly and willfully.” Third, he contends that, as applied to his statements, the statute is unconstitutionally vague.
Reviewing the denial of Phillipos’s motion for acquittal de novo, eschewing credibility judgments, and drawing all reasonable inferences in favor of the verdict, see United States v. Mardirosian, 602 F.3d 1, 7 (1st Cir. 2010), we conclude that a rational jury could have found beyond a reasonable doubt that section 1001 encompassed statements in each count for which Phillipos was convicted and that Phillipos made these statements knowingly and willfully.
We also reject Phillipos’s constitutional vagueness challenge.
Phillipos relies on two separate arguments in contending that the record does not suffice to show that the five false statements at issue are of the type which violate section 1001. We reject both.
1.
Phillipos first argues that the evidence at trial did not suffice to show that his statements were “material,” as section 1001 requires. In rejecting this aspect of Phillipos’s challenge, we base our analysis on our decision in United States v. Mehanna, 735 F.3d 32 (1st Cir. 2014), cert. denied, — U.S. -, 135 S.Ct. 49, 190 L.Ed.2d 29 (2014).
There, a jury had convicted the defendant, Tarek Mehanna, of violating section 1001 by making false statements in connection with an ongoing terrorism investigation to the FBI during informal interviews not unlike the ones that are at issue here. 735 F.3d at 41-42. The statements concerned the whereabouts of one of Me-hanna’s friends, Daniel Maldonado. Id. When FBI agents asked Mehanna when he had last heard from Maldonado, Me-hanna falsely told the agents that he had last heard from Maldonado two weeks earlier and that Maldonado was living in Egypt. Id. at 54. In fact, however, Mehan-na had spoken to Maldonado that week and was aware that Maldonado was in Somalia training for violent jihad. Id
On appeal, Mehanna argued that no reasonable jury could find that his false statements were material, because the agents knew the answers to their questions even before they asked them, such that Mehan-na’s false statements did not in fact mislead them. Id We rejected that argument. Id. at 54-55. We held that to be material, a false statement “need not actually have influenced the governmental function,” id. at 54, but rather need only be of the kind that “could have provoked governmental action,” id (quoting United States v. Sebaggala, 256 F.3d 59, 65 (1st Cir. 2001)). We explained that “the proper inquiry is not whether the tendency to influence bears upon a particular aspect of the investigation but, rather, whether it would bear upon the investigation in the abstract or in the normal course.” Id.
Applying that standard, we then concluded that a jury reasonably could find that Mehanna’s statements about Maldonado “had a natural tendency to influence an FBI investigation into terrorism.” Id. at 55. In setting forth this conclusion, we explained that the record supported a jury’s finding that “the defendant was plainly attempting to obscure both Maldonado’s participation in terrorist endeavors and the telephone call in which he and Maldonado had discussed jihad and terrorist training,” and that “the defendant’s mendacity was undertaken for the purpose of misdirecting the ongoing FBI investigation.” Id.
Given Mehanna, we have little difficulty rejecting Phillipos’s contention that the evidence is insufficient to support Phillipos’s convictions under both of the false statement counts in the indictment. The jury found Phillipos guilty, under both counts of the indictment, of making false statements that are akin to the statements we found material in Mehanna. See, e.g., United States v. Natelli, 527 F.2d 311, 324 (2d Cir. 1975) (recognizing that where a single count of an indictment charges a defendant with multiple false statements, jury can convict provided that it unanimously agrees defendant was guilty of making at least one of the false statements charged); United States v. Duncan, 850 F.2d 1104,
1105 (6th Cir. 1988) (same), overruled on other grounds by Schad v. Arizona, 501 U.S. 624, 111 S.Ct. 2491, 115 L.Ed.2d 555 (1991); United States v. Mangieri, 694 F.2d 1270, 1279-81 (D.C. Cir. 1982) (same); United States v. Jessee, 605 F.2d 430, 431 (9th Cir. 1979) (per curiam) (same).
Specifically, the jury found Phillipos guilty under Count One for stating, on April 20, that, when he returned to the door of Tsarnaev’s dormitory room on April 18 at approximately 10:00 P.M. with Kadyrbayev and Tazhayakov, no one entered the room. And the jury found Philli-pos guilty under Count Two for the following statements on April 25: that Phillipos was not aware of Kadyrbayev or Tazhaya-kov taking anything from Tsarnaev’s dormitory room on the evening of April 18; and that neither he nor Kadyrbayev or Tazhayakov took a backpack from Tsar-naev’s dormitory room on the evening of April 18.
These statements were made in the midst of a federal terrorism investigation. And they provided false information about whether Phillipos and his compatriots entered the bombing suspect’s dormitory room soon after the deadly bombing, went into the suspect’s backpack, and left the room with evidence in tow. Thus, like the statements in Mehanna, these statements by Phillipos could reasonably be deemed to have been intended to obscure the potentially unlawful activities of the defendant’s friends from law enforcement and thereby to frustrate.an ongoing terrorism investigation. See Mehanna, 735 F.3d at 55 (“During the critical interview, the defendant was plainly attempting to obscure both Maldonado’s participation in terrorist endeavors and the telephone call in which he and Maldonado had discussed jihad and terrorist training. The misinformation imparted by the defendant thus had a natural propensity to influence an FBI investigation into terrorist activity.”).
To be sure, other individuals had given law enforcement information about the backpack. Phillipos thus contends that law enforcement already knew the information that his false statements obscured. Mehan-na makes clear, however, that such knowledge on the part of law enforcement is of no moment in determining whether false statements are material. What matters is whether the statement in question would be material to an investigation in the normal course, not whether the statement was actually material to the particular investigation in fact. And, here, a jury could reasonably find that Phillipos’s statements deprived the agents of important corroborating information regarding where, when, and by whom key evidence had been removed.
We recognize that Phillipos contends that Mehanna is distinguishable because the investigation into the Boston Marathon bombing was already complete at the time Phillipos made the false statements. But, as the government rightly notes, it is impossible to conclude that a jury could not reasonably find otherwise. Even though Tsarnaev had been apprehended at the time of Phillipos’s interviews, the government still had an interest in continuing to investigate the matter, given the significant public safety interest in determining who might have been involved in the bombing and the government’s need to develop the case for prosecution. Thus, Phillipos’s materiality-based sufficiency challenge fails.
2.
Phillipos also contends that his motion for acquittal was wrongly denied — Mehan-na’s materiality holding notwithstanding— because none of the false statements at issue are “the type of statements and con
duct that [C]ongress intended to criminalize.” To support this contention, Phillipos relies on a line of cases that had established, for a time, what was known as the “exculpatory no” doctrine.
When the cases in that line were decided, the version of section 1001 that was in place prohibited the making of “any false, fictitious or fraudulent statements” in certain government matters. See Brogan v. United States, 522 U.S. 398, 400, 118 S.Ct. 805, 139 L.Ed.2d 830 (1998) (quoting 18 U.S.C. § 1001 (1988 ed.)). The “exculpatory no” doctrine interpreted that version of section 1001, despite its seemingly all-encompassing sweep, not to apply to a statement that a defendant made to a law enforcement officer during an informal interview in which the defendant simply denied engaging in wrongdoing. See, e.g., United States v. Chevoor, 526 F.2d 178, 182 (1st Cir. 1975) (holding that statements that “fall within the ‘exculpatory no’ category of responses ... are outside the scope of ‘statements’ within the meaning of the statute”).
Phillipos contends that “Chevoor’s scenario is nearly identical” to his own, because, like the defendant in. Chevoor, Phil-lipos “did not fabricate a misleading story ... or send the [federal] agents on a wild goose chase.” Instead, he merely “gave negative, oral responses to the questioning.” As the government points out, however, the “exculpatory no” doctrine that we adopted in Chevoor was overturned by the Supreme Court in Brogan, 522 U.S. 398, 118 S.Ct. 805. In Brogan, the Supreme Court held that, by its plain terms, the earlier version of section 1001 “cover[ed] ‘any’ false statement — that is, a false statement ‘of whatever kind,’ ” and “[t]he word ‘no’ in response to a question assuredly makes a ‘statement.’ ” Id. at 400-01, 118 S.Ct. 805 (citation omitted). As such, the Court held that the plain text of the statute as it then existed applied to “exculpatory no” statements, even though, as Justice Ginsburg observed, “[i]t is doubtful Congress intended § 1001 to cast so large a net.” Id at 412, 118 S.Ct. 805 (Ginsburg, J., concurring in the judgment). .
Phillipos’s only answer is to suggest that Brogan overruled only “portions” of Chev-oor, and that Chevoor’s holding regarding the “exculpatory no” doctrine remains binding regardless. This contention, of course, has no merit, given that Chevoor’s construction of the- statute was explicitly ■ rejected. And Phillipos develops no argument as to how some version of the “exculpatory no” lives on in the current version of section 1001.
In any event, for the same reasons that a jury could reasonably find material the statements by Phillipos that we discussed above about who did or did not enter the lead suspect’s dormitory room and take evidence from his backpack shortly after the bombing occurred, a jury also could
reasonably reject Phillipos’s contention that such statements were not meant “to fabricate a story or send agents on a wild goose chase.” See Mehanna, 735 F.3d at 55 (“[W]here a defendant’s statements are intended to misdirect government investigators, they may satisfy the materiality requirement of section 1001 even if they stand no chance of accomplishing their objective.”). Thus, this sufficiency challenge fails as well.
Phillipos next trains his focus on the fact that section 1001 applies to only those false statements and representations that an individual makes knowingly and willfully. 18 U.S.C. § 1001(a). Phillipos contends that, even if he violated the statute, the government failed to present sufficient evidence that he did so with the requisite mens rea. We disagree.
The Supreme Court has made clear that “in order to establish a ‘willful’ violation of a statute, ‘the Government must prove that the defendant acted with knowledge that his conduct was unlawful.’ ” Bryan v. United States, 524 U.S. 184, 191-92, 118 S.Ct. 1939, 141 L.Ed.2d 197 (quoting Ratzlaf v. United States, 510 U.S. 135, 137, 114 S.Ct. 655, 126 L.Ed.2d 615 (1994)). But “[willfulness can rarely be proven by direct evidence, since it is a state of mind.” United States v. Bank of New Eng., N.A., 821 F.2d 844, 854 (1st Cir. 1987). As such, willfulness is “usually established by drawing reasonable inferences from available facts.” Id.
With respect to the false statements in Count Two, which Phillipos made during the April 25 interview, there would have been no need for the jury to infer much. The interviewing agent testified at trial that he affirmatively told Phillipos that Phillipos could be prosecuted for making false statements. This direct evidence suffices to support a jury’s finding that section 1001’s willfulness requirement had been met.
The government did not present similarly direct evidence with respect to the statements at issue in Count One, which Phillipos made during the April 20 interview. But, the government notes, by April 20 — five days after the Boston Marathon bombing occurred — Phillipos, along with his friends, had become the focus of intense law enforcement interest due to their relationship with Tsarnaev. And, Phillipos had already been interviewed by law enforcement the previous day, on April 19.
A jury could reasonably infer that someone who potentially had information about the removal and destruction of evidence in a historic terrorism investigation into a deadly attack on a symbolic event, and who was interviewed by federal law enforcement agents twice, would know that it was unlawful to make false statements to investigators about what he knew. The reasonableness of this inference is further bolstered by the fact that, on April 20, Phillipos’s interviewer told him that “now [was] the time to tell [the government]” what he knew. The interviewer also asked whether “this w^s the story he want[ed] to go with” and gave Phillipos the opportunity to “correct” it. We thus reject this challenge.
C.
Finally, Phillipos contends that section 1001 is unconstitutionally vague as applied to the five false statements that he was convicted of making. He argues that, by virtue of the express materiality requirement that section 1001 now contains, it is not clear which statements fall under section 1001 and which do not. But this argument, too, is unavailing.
A criminal statute is void for vagueness only if it “fails to provide a person of ordinary intelligence fair notice of what is prohibited, or is so standardless that it authorizes or encourages seriously discriminatory enforcement.” United States v. Williams, 553 U.S. 285, 304, 128 S.Ct. 1830, 170 L.Ed.2d 650 (2008). Applying section 1001’s materiality standard to Phillipos’s statements raises no such concerns. A criminal statute is not unconstitutionally vague merely because it sets forth a standard for determining liability that is not mathematically precise. And here, the challenged standard — “materiality”—is a familiar one. Indeed, “[i]ts use in the context of false statements to public officials goes back as far as Lord Coke” in 1680. Kungys v. United States, 485 U.S. 759, 769, 108 S.Ct. 1537, 99 L.Ed.2d 839 (1987).
Moreover, Mehanna laid out a test for determining when statements are material under section 1001 that used the same language as the test that had been set forth decades before in Kungys, which itself tracked the “uniform understanding of the ‘materiality’ concept” that “federal courts have long displayed.” See Kungys, 485 U.S. at 770, 108 S.Ct. 1537. Under that standard, as we have explained, the evidence was more than sufficient to permit a reasonable jury to find the materiality requirement was satisfied as to each count. Thus, we reject Phillipos’s as-applied, constitutional vagueness challenge.
IV.
In light of the foregoing, we affirm.