The Divergence Borel-Cantelli Lemma revisited

Beresnevich, Victor; Velani, Sanju

doi:10.48550/arxiv.2103.12200

Cited by 3 publications

(20 citation statements)

References 28 publications

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“…(v) The conclusion of Theorem 1 is close to Rudnick's upper bound (2) in cases where E N ≈ N 2 and thus C N ≈ N 2 (only improving the error N ε to errors of logarithmic order). Rudnick's result also meets with that of Theorem 1 when E N ≈ N 3 and C N ≈ N. However, in the intermediate range when E N is around N δ for some δ ∈ (2, 3), the estimate ( 2) is significantly weaker than (5), except in the "random" case where all possible differences a m − a n have a similar number of representations so that E N ≈ N 4 /C N . Examples of such sequences include the Piatetski-Shapiro integers as well as the sequence of (log x) A smooth numbers n ≤ x where we expect C N ≈ N β A for some β A in the open interval (1, 2); see [3] for more information.…”

Section: Introduction and Statement Of Resultssupporting

confidence: 57%

“…It is known that in metric Diophantine approximation one does not need full stochastic independence, but that it is sufficient to establish "quasi-independence on average" (cf. [4,5,6] as well as Lemma 4 below). The key for this is to control the measure of the overlaps S m ∩ S n for m = n. In Rudnick's paper [34] this is done by a direct application of L 2 methods, which essentially gives the overlap estimate…”

Section: Proof Of Theorem 1 Part 2: Upper Boundmentioning

confidence: 76%

“…Examples of such sequences include the Piatetski-Shapiro integers as well as the sequence of (log x) A smooth numbers n ≤ x where we expect C N ≈ N β A for some β A in the open interval (1, 2); see [3] for more information. (vi) From the viewpoint of metric number theory, an upper bound such as (5) is usually much more difficult to establish than a lower bound such as (6). This is because the upper bound (5) uses the second rather than the first Borel-Cantelli lemma, which only holds under some additional "stochastic independence" requirement.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…As usual in metric number theory, the "divergence" part is much more difficult than the "convergence" part, since the divergence part of the Borel-Cantelli lemma requires some form of stochastic independence, while the convergence part holds unconditionally; see [5] for a detailed discussion of this issue. In Section 4.1, we will prove the upper bound of Theorem 1 which depends on the size of a N , and sketch the argument leading to a bound that is independent of the size of a N .…”

Section: Proof Of Theorem 1 Part 2: Upper Boundmentioning

confidence: 99%

See 3 more Smart Citations

Difference sets and the metric theory of small gaps

Aistleitner¹,

El-Baz²,

Munsch³

2021

Preprint

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Let (a n ) n≥1 be a sequence of distinct positive integers. In a recent paper Rudnick established asymptotic upper bounds for the minimal gaps of {a n α mod 1, 1 ≤ n ≤ N } as N → ∞, valid for Lebesgue-almost all α and formulated in terms of the additive energy of {a 1 , . . . , a N }. In the present paper we argue that the metric theory of minimal gaps of such sequences is not controlled by the additive energy, but rather by the cardinality of the difference set of {a 1 , . . . , a N }. We establish a (complicated) sharp convergence/divergence test for the typical asymptotic order of the minimal gap, and prove (slightly weaker) general upper and lower bounds which allow for a direct application. A major input for these results comes from the recent proof of the Duffin-Schaeffer conjecture by Koukoulopoulos and Maynard. We show that our methods give very precise results for slowly growing sequences whose difference set has relatively high density, such as the primes or the squares. Furthermore, we improve a metric result of Blomer, Bourgain, Rudnick and Radziwi l l on the order of the minimal gap in the eigenvalue spectrum of a rectangular billiard.

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Section: Introduction and Statement Of Resultssupporting

confidence: 57%

Section: Proof Of Theorem 1 Part 2: Upper Boundmentioning

confidence: 76%

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Section: Proof Of Theorem 1 Part 2: Upper Boundmentioning

confidence: 99%

See 2 more Smart Citations

Difference sets and the metric theory of small gaps

Aistleitner¹,

El-Baz²,

Munsch³

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…A foundational result in Diophantine approximation is Dirichlet's approximation theorem, which asserts that for every real number α there are infinitely many coprime solutions (p, q) to the inequality (1) α − p q < 1 q 2 . It is well-known that this result is optimal up to constant factors for numbers α whose partial quotients in the continued fraction representation are bounded (so-called badly approximable numbers).…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

On the metric theory of approximations by reduced fractions: a quantitative Koukoulopoulos-Maynard theorem

Aistleitner¹,

Borda²,

Manuel³

2022

Preprint

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Let ψ : N → [0, 1/2] be given. The Duffin-Schaeffer conjecture, recently resolved by Koukoulopoulos and Maynard, asserts that for almost all reals α there are infinitely many coprime solutions (p, q) to the inequality |α− p/q| < ψ(q)/q, provided that the series ∞ q=1 ϕ(q)ψ(q)/q is divergent. In the present paper, we establish a quantitative version of this result, by showing that for almost all α the number of coprime solutions (p, q), subject to q ≤ Q, is of asymptotic order Q q=1 2ϕ(q)ψ(q)/q. The proof relies on the method of GCD graphs as invented by Koukoulopoulos and Maynard, together with a refined overlap estimate coming from sieve theory, and number-theoretic input on the "anatomy of integers". The key phenomenon is that the system of approximation sets exhibits "asymptotic independence on average" as the total mass of the set system increases.

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Independence inheritance and Diophantine approximation for systems of linear forms

Allen¹,

Ramírez²

2021

Preprint

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The classical Khintchine-Groshev theorem is a generalization of Khintchine's theorem on simultaneous Diophantine approximation, from approximation of points in R m to approximation of systems of linear forms in R nm . In this paper, we present an inhomogeneous version of the Khintchine-Groshev theorem which does not carry a monotonicity assumption when nm > 2. Our results bring the inhomogeneous theory almost in line with the homogeneous theory, where it is known by a result of Beresnevich and Velani (2010) that monotonicity is not required when nm > 1. That result resolved a conjecture of Beresnevich, Bernik, Dodson, and Velani (2009), and our work resolves almost every case of the natural inhomogeneous generalization of that conjecture. Regarding the two cases where nm = 2, we are able to remove monotonicity by assuming extra divergence of a measure sum, akin to a linear forms version of the Duffin-Schaeffer conjecture. When nm = 1 it is known by work of Duffin and Schaeffer (1941) that the monotonicity assumption cannot be dropped.The key new result is an independence inheritance phenomenon; the underlying idea is that the sets involved in the ((n + k) × m)-dimensional Khintchine-Groshev theorem (k ≥ 0) are always k-levels more probabilistically independent than the sets involved the (n × m)-dimensional theorem. Hence, it is shown that Khintchine's theorem itself underpins the Khintchine-Groshev theory.

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The Divergence Borel-Cantelli Lemma revisited

Cited by 3 publications

References 28 publications

Difference sets and the metric theory of small gaps

Difference sets and the metric theory of small gaps

On the metric theory of approximations by reduced fractions: a quantitative Koukoulopoulos-Maynard theorem

Independence inheritance and Diophantine approximation for systems of linear forms

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