SUMMARY ORDER
Defendant-Appellant Ben F. Andrews (“Andrews” or “Defendant”) appeals from a judgment of conviction entered on June 9, 2005, in the United States District Court for the District of Connecticut (Burns, /.). Following a jury trial, Andrews was convicted of nine counts, including bribery, fraud, money-laundering, and false-statement charges, for his role in former Treasurer of the State of Connecticut Paul Silvester’s investment of state pension funds with an investment firm named Landmark. On appeal Andrews argues that there was insufficient evidence to support the corruption and false statement charges; that a new trial is warranted because the district court abused its discretion in various evidentiary rulings; that the district court erred in not requiring the Government to grant immunity to two witnesses who had invoked their Fifth Amendment privilege against self-incrimination; and that the district court violated Andrews’ Sixth Amendment right to counsel when it placed restrictions on communication between him and his counsel. We consider Andrews’ Sixth Amendment claim in the opinion accompanying this summary order and address his remaining claims below. We assume the parties’ familiarity with the facts, the procedural history, and the specific issues on appeal.
Andrews first argues that there was insufficient evidence to establish that Andrews knew that Silvester based his decisions to invest in Landmark on improper motives. In assessing a challenge to the sufficiency of the evidence underlying a conviction, we review the trial evidence in the light most favorable to the government and draw all permissible inferences in the government’s favor. United States v. Jackson, 335 F.3d 170, 180 (2d Cir.2003). The jury was entitled to credit the testimony that Andrews knew and intended that some of his “finder’s fee” would be kicked back to Silvester and that neither Andrews nor Christopher Sack did any meaningful work on the Landmark deal. On the basis of this evidence, a rational jury could infer that it was beyond a reasonable doubt that Andrews must have understood that Silvester was motivated to do the Landmark deal in part to effectuate an illicit enrichment of himself and his friends. See United States v. Payton, 159 F.3d 49, 56 (2d Cir.1998) (“The ultimate question is not whether we believe the evidence adduced at trial established defendant’s guilt beyond a reasonable doubt, but whether any rational trier of fact could so find.”).
Andrews also argues that there was not sufficient evidence to support his conviction for making a false statement to federal agents in violation of 18 U.S.C. § 1001, because of ambiguity in the indictment and in the questioning during which the allegedly false statement was made. This Court has recognized that whether a defendant has “knowingly and willfully ... ma[de] any materially false, fictitious or fraudulent statements or representations” under § 1001 is governed by the same legal standards as whether a defendant “willfully subscribes as true any material matter which he does not believe to be true” in violation of the perjury statute, 18 U.S.C. § 1621. See United States v. Mandanici, 729 F.2d 914, 921 (2d Cir.1984) (citing Bronston v. United States, 409 U.S. 352, 93 S.Ct. 595, 34 L.Ed.2d 568 (1973)). Generally the meaning and truthfulness of a defendant’s statement is a question of *628fact for the jury. United States v. Lighte, 782 F.2d 367, 372 (2d Cir.1986). A reviewing court can override a properly charged jury verdict as a matter of law only if the question to which the allegedly false statement was in answer “is so vague as to be ‘fundamentally ambiguous.’ ” Id. at 375; see also United States v. Markiewicz, 978 F.2d 786, 808-09 (2d Cir.1992). A question is fundamentally ambiguous when, taking the context of the line of questioning as a whole into account, it “ ‘is not a phrase with a meaning about which men of ordinary intellect could agree, nor one which could be used with mutual understanding by a questioner and answerer unless it were defined at the time it were sought and offered into testimony.’ ” Lighte, 782 F.2d at 375 (quoting United States v. Lattimore, 127 F.Supp. 405, 410 (D.D.C.1955), aff'd by an equally divided court, 232 F.2d 334 (D.C.Cir.)).
Count 12 of the indictment charged the Defendant with making a false statement when he told an FBI agent, that he “had contacted Christopher Stack to become involved in an investment deal by the State of Connecticut with Fund A because he felt that Stack would be helpful in finalizing the deal with the Treasurer, but that he (ANDREWS) had not discussed this arrangement with Stack in front of Paul J. Silvester____” (emphasis added). Similarly, Special Agent Joseph McTague, one of the FBI agents who questioned Andrews, testified that he had asked Andrews whether he had “discussed this fee splitting arrangement with Mr. Silvester,” and that Andrews told the agents that he and Silvester “had discussed the Landmark deal but not [Andrews and Sack’s] fee splitting arrangement.” Andrews admits that if “this arrangement” means the fact that Andrews would split his fee with Stack, then Andrews’ answer was untruthful. But Andrews claims that the term “arrangement” is ambiguous and that he believed when he was speaking to the agents that “this arrangement” meant “the arrangements by which Rogers & Wells received consulting payments from Landmark, which it distributed to Mr. Andrews and KCATS.”
There is no fundamental ambiguity that would justify overturning the conviction. It is apparent from the context of both the indictment and Agent McTague’s testimony that “this arrangement” referred, without any significant ambiguity, to the existence of the fee-splitting agreement. See United States v. Carey, 152 F.Supp.2d 415, 428-29 (S.D.N.Y.2001) (distinguishing fundamentally ambiguous phrases which “virtually require an explicit definition” from “the meanings of pronouns such as ‘this,’ ‘that,’ ‘those,’ etc.,” which “can generally be elucidated by implication from the surrounding text”). It is significant that the Defendant cannot discuss the interpretation he favors without adding an “s” onto the word “arrangement.” Ordinary speakers would generally distinguish between “the fee splitting arrangement” itself and the “arrangements ” made to carry out the fee splitting.
Moreover, Andrews’ explanation, as testified to by Special Agent McTague, for why he did not tell Silvester about the fee splitting arrangement is far more logical if Andrews understood “arrangement” in the broader sense. And this too provides substantial evidence that Andrews did knowingly make a false statement. See Lighte, 782 F.2d at 375 (explaining that even if a question is ambiguous, “if the response given was false as the defendant understood the question, his conviction is not invalidated by the fact that his answer to the question might generate a number of different interpretations.”) (quoting United States v. Williams, 552 F.2d 226, 229 (8th Cir.1977)).
Andrews’ third argument on appeal is that the district court erred in refusing to *629inquire before the end of Andrews’ direct examination about the specific evidence of misconduct the government intended to introduce pursuant to Federal Rule of Evidence 608(b) during cross examination. Reviewing the district court’s evidentiary decisions for an abuse of discretion, as we must, see, e.g., United States v. Yousef, 327 F.3d 56, 156 (2d Cir.2003), we do not find any here
Rule 608(b) allows a witness to be cross-examined about specific instances of conduct “in the discretion of the court” if these are probative of his truthfulness. There is no support for Andrews’ novel contention that Rule 608(b)’s reference to the “discretion of the court” requires, as a matter of law, that the district court issue a ruling in advance of the cross examination about what impeaching material will be allowed. To the contrary, the district court has particularly broad discretion to determine the timing and procedure for its evidentiary rulings. See Fed. R. Evid. 611, Advisory Committee Notes (“[Spelling out detailed rules to govern the mode and order of interrogating witnesses and presenting evidence is neither desirable nor feasible. The ultimate responsibility for the effective working of the adversary system rests with the judge.”).
To the extent that Andrews is also arguing the merits of the evidentiary ruling, i.e., that the district court abused its discretion in allowing Andrews to be cross examined about specific instances in which he made false statements, including false sworn statements, this argument too fails. This Court has held that it can be appropriate to introduce false statements, especially false sworn statements, under Rule 608(b)(1) to shed light on a witness’ credibility. See e.g., United States v. Jones, 900 F.2d 512, 520-521 (2d Cir.1990). Andrews’ credibility was a crucial issue at trial. Although evidence may be excluded if its “probative value is substantially outweighed by the danger of unfair prejudice,” Fed. R. Evid. 403, the district court was well within its discretion in finding that any danger of unfair prejudice from the allowed cross examination did not substantially outweigh its probative value. Cf. United States v. Whitmore, 359 F.3d 609, 619 (D.C.Cir.2004) (“Nothing could be more probative of a witness’s character for untruthfulness than evidence that the witness has previously lied under oath.”).
Andrews’ final claim is that the district court erred in refusing to “cure” two instances of admitted hearsay evidence by compelling the government to grant immunity to two potential witnesses who had invoked their Fifth Amendment privilege. Although Andrews’ argument may imply otherwise, whether the district court erred in not compelling immunization and whether inadmissible hearsay was admitted are two independent inquiries.
We consider first the district court’s refusal to compel the government to grant immunity, which we review for abuse of discretion, United States v. Ebbers, 458 F.3d 110, 118 (2d Cir.2006). We have held that “[ajbsent extraordinary circumstances, the Due Process Clause imposes no requirement that defense witness immunity be ordered whenever it seems fair to grant it.” United States v. Diaz, 176 F.3d 52, 115 (2d Cir.1999) (quoting Blissett v. Lefevre, 924 F.2d 434, 441 (2d Cir.1991) (internal quotation marks omitted)). We use a three-part test to determine whether there are exceptional circumstances:
First, the district court must find that the government, through its own overreaching, has forced the witness to invoke the Fifth Amendment or, that the government has engaged in discriminatory use of grants of immunity to gain a tactical advantage; second, the witness’ testimony must be material, exculpatory and not cumulative; and third, the de*630fendant [must have] no other source to obtain the evidence.
Id. The defendant bears the burden of showing that each of the three elements is met. Id.
The district court did not abuse its discretion in concluding that Andrews had failed to meet his burden. First, it is not sufficient simply to show that the prosecution granted immunity to its own witnesses and not to the defendants’; the defendant must prove discriminatory use of immunity for the purpose of gaining a tactical advantage. Blissett, 924 F.2d at 442; see also United States v. Todaro, 744 F.2d 5, 10 (2d Cir.1984) (“The number of witnesses immunized by the Government, without more, would not support a finding of this type of misconduct.”). Andrews — who bases his argument solely on the fact that some Government witnesses were granted immunity — provided no evidence that the Government’s decision not to grant immunity to Jerome Wilson and Stanley Alfeld was improper. Second, as Andrews admits in his brief, no one knows what Wilfred and Alfeld would have testified to since they refused to comment on the matter. Andrews’ speculation that Wilson and Alfeld would have testified in Andrews’ favor is not sufficient to prove that their testimony would have been exculpatory.
Andrews’ claim that two instances of hearsay relating to Wilson and Alfeld were admitted in violation of the Confrontation Clause is also without merit. Hearsay admissibility determinations are also reviewed for abuse of discretion, United States v. Tocco, 135 F.3d 116, 127 (2d Cir.1998), and subject to harmless error analysis, United States v. Orena, 32 F.3d 704, 711 (2d Cir.1994). The first instance of alleged hearsay occurred during the direct examination of Silvester, when Silvester stated, before he was interrupted by an objection from the defense counsel, “The Landmark deal is — what I did there is I invested money in this organization and I asked them to pay — I—through an intermediary, I asked them to pay.” The district court did not abuse its discretion in admitting this testimony. Andrews claims that Silvester’s answer amounted to testimony about “conversations between Wilson and Alfeld.” But as is apparent from Silvester’s subsequent uninterrupted answer,1 Silvester was not testifying to the contents of conversations between Wilson and Alfeld, but rather to what he told Wilson.2
Andrews also objects, on the grounds that it was not properly authenticated, to the entry into evidence of a draft letter that Wilson allegedly prepared for Landmark. Authentication requires only “evidence sufficient to support a finding that the matter in question is what its proponent claims.” Fed. R. Evid. 901(a). Andrews testified that the content of the letter reflected a discussion that the he had with Wilson and that the letter was “[Wilson’s] words.” It was within the district court’s “broad discretion,” United States v. Tropeano, 252 F.3d 653, 661 (2d Cir.2001), to conclude that this testimony was sufficient to show that the letter was what the Government claimed — a letter *631that reflected discussions Wilson had with Andrews about the Landmark deal. Hence admission of the letter was not error. Moreover, even if there had been error, it was harmless. The letter only confirmed Andrews’ testimony about the work he did for Landmark, and Andrews repeatedly admitted that the content of the letter accurately reflected at least part of the conversation he had with Wilson.
Accordingly, for the foregoing reasons we AFFIRM the judgment of the district court.
. When Silvester was allowed to finish his answer, he stated, "I made a decision to invest in Landmark, but before doing so, I went to a gentleman by the name of Jerry Wilson and asked him to speak to Landmark and arrange for Mr. Andrews and Mr. Stack to be paid a finder’s fee.”
. Out-of-court statements made by Silvester regarding the Landmark deal are admissible under Federal Rule of Evidence 801(d)(2)(E), which excludes from the definition of hearsay statements made by a co-conspirator during the course, and in furtherance, of a conspiracy. See United States v. Tellier, 83 F.3d 578, 580 (2d Cir.1996). Andrews does not argue to the contrary.