TY - JOUR

T1 - Equal sums in random sets and the concentration of divisors

AU - Ford, Kevin

AU - Green, Ben

AU - Koukoulopoulos, Dimitris

N1 - Funding Information:
This collaboration began at the MSRI program on Analytic Number Theory, which took place in the first half of 2017 and which was supported by the National Science Foundation under Grant No. DMS-1440140. All three authors are grateful to MSRI for allowing us the opportunity to work together. The project was completed during a visit of KF and DK to Oxford in the first half of 2019. Both authors are grateful to the University of Oxford for its hospitality. KF is supported by the National Science Foundation Grants DMS-1501982 and DMS-1802139. In addition, his stay at Oxford in early 2019 was supported by a Visiting Fellowship at Magdalen College Oxford. BG is supported by a Simons Investigator Grant, which also funded DK’s visit to Oxford. DK is also supported by the Courtois Chair II in fundamental research, by the Natural Sciences and Engineering Research Council of Canada (RGPIN-2018-05699) and by the Fonds de recherche du Québec - Nature et technologies (2019-PR-256442 and 2022-PR-300951).
Funding Information:
This collaboration began at the MSRI program on Analytic Number Theory, which took place in the first half of 2017 and which was supported by the National Science Foundation under Grant No. DMS-1440140. All three authors are grateful to MSRI for allowing us the opportunity to work together. The project was completed during a visit of KF and DK to Oxford in the first half of 2019. Both authors are grateful to the University of Oxford for its hospitality. KF is supported by the National Science Foundation Grants DMS-1501982 and DMS-1802139. In addition, his stay at Oxford in early 2019 was supported by a Visiting Fellowship at Magdalen College Oxford. BG is supported by a Simons Investigator Grant, which also funded DK’s visit to Oxford. DK is also supported by the Courtois Chair II in fundamental research, by the Natural Sciences and Engineering Research Council of Canada (RGPIN-2018-05699) and by the Fonds de recherche du Québec - Nature et technologies (2019-PR-256442 and 2022-PR-300951).
Publisher Copyright:
© 2023, The Author(s).

PY - 2023/6

Y1 - 2023/6

N2 - We study the extent to which divisors of a typical integer n are concentrated. In particular, defining Δ (n) : = max t# { d| n, log d∈ [t, t+ 1] } , we show that Δ (n) ⩾ (log log n) 0.35332277 … for almost all n, a bound we believe to be sharp. This disproves a conjecture of Maier and Tenenbaum. We also prove analogs for the concentration of divisors of a random permutation and of a random polynomial over a finite field. Most of the paper is devoted to a study of the following much more combinatorial problem of independent interest. Pick a random set A⊂ N by selecting i to lie in A with probability 1/i. What is the supremum of all exponents βk such that, almost surely as D→ ∞, some integer is the sum of elements of A∩[Dβk,D] in k different ways? We characterise βk as the solution to a certain optimisation problem over measures on the discrete cube { 0 , 1 } k, and obtain lower bounds for βk which we believe to be asymptotically sharp.

AB - We study the extent to which divisors of a typical integer n are concentrated. In particular, defining Δ (n) : = max t# { d| n, log d∈ [t, t+ 1] } , we show that Δ (n) ⩾ (log log n) 0.35332277 … for almost all n, a bound we believe to be sharp. This disproves a conjecture of Maier and Tenenbaum. We also prove analogs for the concentration of divisors of a random permutation and of a random polynomial over a finite field. Most of the paper is devoted to a study of the following much more combinatorial problem of independent interest. Pick a random set A⊂ N by selecting i to lie in A with probability 1/i. What is the supremum of all exponents βk such that, almost surely as D→ ∞, some integer is the sum of elements of A∩[Dβk,D] in k different ways? We characterise βk as the solution to a certain optimisation problem over measures on the discrete cube { 0 , 1 } k, and obtain lower bounds for βk which we believe to be asymptotically sharp.

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U2 - 10.1007/s00222-022-01177-y

DO - 10.1007/s00222-022-01177-y

M3 - Article

AN - SCOPUS:85151275045

SN - 0020-9910

VL - 232

SP - 1027

EP - 1160

JO - Inventiones Mathematicae

JF - Inventiones Mathematicae

IS - 3

ER -