On the maximal measure of a spherical set avoiding solutions to x + y + z = 0

Ákos Dúcz

On the maximal measure of a spherical set avoiding solutions to x + y + z = 0

Combinatorics 2026-07-06 v1

Authors: Ákos Dúcz

Abstract

We prove that the maximal normalized surface measure of a spherical set in d dimensions avoiding solutions to x + y + z = 0 approaches 1/2 as d goes to infinity. This gives a partial answer to a question of Bukh, who conjectured 1/2 to be the optimal bound for all d >= 3.

Keywords

spherical sets zero-sum-free maximal measure asymptotic density Bukh's conjecture

Cite

@article{arxiv.2607.05083,
  title  = {On the maximal measure of a spherical set avoiding solutions to x + y + z = 0},
  author = {Ákos Dúcz},
  journal= {arXiv preprint arXiv:2607.05083},
  year   = {2026}
}

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