Small Gál sums and applications

Bretèche, Régis de la; Munsch, Marc; Tenenbaum, Gérald

doi:10.1112/jlms.12378

Cited by 10 publications

(12 citation statements)

References 34 publications

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“…This is joint work with Kerr and Shparlinski [8]. Finally, we remark that de la Bretèche and Munsch [2] and de la Bretèche et al [3] have improved our result building on this idea.…”

mentioning

confidence: 54%

Distribution of Integers With Prescribed Structure and Applications

Yau

2020

Bull. Aust. Math. Soc.

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“…This is joint work with Kerr and Shparlinski [8]. Finally, we remark that de la Bretèche and Munsch [2] and de la Bretèche et al [3] have improved our result building on this idea.…”

mentioning

confidence: 54%

Distribution of Integers With Prescribed Structure and Applications

Yau

2020

Bull. Aust. Math. Soc.

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“…This implies that N 40/21+2ε (E * N ) 13/15 ≪ N 4− δ 2 +2ε , which is ≪ N 4−ε/4 for a sufficiently small ε. Thus for every dyadic interval or in another words every fixed u, we have an upper bound of order N 4−ε/4 for the total contribution of z m , z n with z m , z n ≥ N 1.01 to (13).…”

Section: Proof Of Theorem 1 Part 3: Lattice Point Counting Via the Ri...mentioning

confidence: 96%

“…, z M } be the multi-set of all absolute differences {|x m − x n | : 1 ≤ m, n ≤ N, m = n}, meaning that we allow repetitions in the definition. For a given u ≤ 2rN , we wish to estimate (13) 1≤m,n≤M,…”

Section: Proof Of Theorem 1 Part 3: Lattice Point Counting Via the Ri...mentioning

confidence: 99%

A pair correlation problem, and counting lattice points with the zeta function

Aistleitner¹,

El-Baz²,

Munsch³

2020

Preprint

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The pair correlation is a localized statistic for sequences in the unit interval. Pseudo-random behavior with respect to this statistic is called Poissonian behavior. The metric theory of pair correlations of sequences of the form (anα) n≥1 has been pioneered by Rudnick, Sarnak and Zaharescu. Here α is a real parameter, and (an) n≥1 is an integer sequence, often of arithmetic origin. Recently, a general framework was developed which gives criteria for Poissonian pair correlation of such sequences for almost every real number α, in terms of the additive energy of the integer sequence (an) n≥1 . In the present paper we develop a similar framework for the case when (an) n≥1 is a sequence of reals rather than integers, thereby pursuing a line of research which was recently initiated by Rudnick and Technau. As an application of our method, we prove that for every real number θ > 1, the sequence (n θ α) n≥1 has Poissonian pair correlation for almost all α ∈ R.

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“…We do not see any reason why the particular set M, arising from the multiplication table problem, should not yield the desired scaling factor, but we have not been able to prove anything in this direction. The estimate (14) seems to be even more delicate, as it does not seem that controlling the pairwise overlaps is sufficient to obtain such a result, but rather the overlaps between more than two sets would need to be controlled. We note in conclusion that there do exist results on metric Diophantine approximation with two restricted variables, most prominently in the work of Harman [22,23,24,25].…”

Section: Proof Of Theoremmentioning

confidence: 99%

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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Small Gál sums and applications

Cited by 10 publications

References 34 publications

Distribution of Integers With Prescribed Structure and Applications

Distribution of Integers With Prescribed Structure and Applications

A pair correlation problem, and counting lattice points with the zeta function

Difference sets and the metric theory of small gaps

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