TY - JOUR

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

AU - Ford, Kevin

AU - Green, Ben

AU - Koukoulopoulos, Dimitris

N1 - 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 -