Journal article
Equal sums in random sets and the concentration of divisors
- Abstract:
- We study the extent to which divisors of a typical integer n are concentrated. In particular, defining Ξ(π):=maxπ‘#{π|π,logπβ[π‘,π‘+1]}, we show that Ξ(π)β©Ύ(loglogπ)0.35332277β¦ 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 πββ by selecting i to lie in π with probability 1/i. What is the supremum of all exponents π½π such that, almost surely as π·ββ, some integer is the sum of elements of πβ©[π·π½π,π·] in k different ways? We characterise π½π as the solution to a certain optimisation problem over measures on the discrete cube {0,1}π, and obtain lower bounds for π½π which we believe to be asymptotically sharp.
- Publication status:
- Published
- Peer review status:
- Peer reviewed
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(Preview, Version of record, pdf, 1.4MB, Terms of use)
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- Publisher copy:
- 10.1007/s00222-022-01177-y
Authors
- Publisher:
- Springer
- Journal:
- Inventiones Mathematicae More from this journal
- Volume:
- 232
- Issue:
- 3
- Pages:
- 1027-1160
- Publication date:
- 2023-03-29
- Acceptance date:
- 2022-12-09
- DOI:
- EISSN:
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1432-1297
- ISSN:
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0020-9910
- Language:
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English
- Keywords:
- Pubs id:
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1039004
- Local pid:
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pubs:1039004
- Deposit date:
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2023-03-03
- ARK identifier:
Terms of use
- Copyright holder:
- Ford et al.
- Copyright date:
- 2023
- Rights statement:
- Β© The Author(s) 2023. Open Access. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the articleβs Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the articleβs Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
- Licence:
- CC Attribution (CC BY)
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