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OP-11912Status unassessedNumber theory

Differences of signed representatives of the first n integers

Differences of signed representatives of the first n integers. Current research status is unassessed. The dated source poses this question; inclusion does not certify that it remains open today.

MSC 11 · Number theory

Editorial MSC2020 mapping of Differences of signed representatives of the first n integers. https://ems.press/content/serial-article-files/52435#page=56 Not an author-supplied code.

Source question · editorial summaryAs of 2025; current status unassessedSource & evidence ↓
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Problem statement

Let A⊂{−n,…,n}∖{0}A\subset\{-n,\ldots,n\}\setminus\{0\} choose exactly one of k,−kk,-k for each 1≤k≤n1\le k\le n. Must ∣(A−A)∩[1,n]∣=(1−o(1))n|(A-A)\cap[1,n]|=(1-o(1))n? More strongly, for each B⊂[n]B\subset[n] with ∣B∣≥εn|B|\ge\varepsilon n, must ∑a,b∈A1B(a−b)≫εn2\sum_{a,b\in A}1_B(a-b)\gg_\varepsilon n^2?

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References
  1. [1]Problem-session contributors; compiled by Thomas F. Bloom. Oberwolfach 2025 — Analytic Number Theory Problem Session (2025).; Differences of signed representatives of the first n integers linkCC BY-SA 4.0; source contributors, compiled by Thomas F. Bloom. These attributed adaptations are available under the same license.

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