PreprintCC BY 4.0arXiv
A negative answer to the Erdős-Sárkőzy question
Abstract
For a finite set A of positive integers, let H(A) be its set of subset sums, and let g3(n) be the least N for which some n-element set A⊆{1,…,N} has H(A) free of nonconstant three-term arithmetic progressions. Erdős and Sárkőzy asked whether g3(n)≫ 3n. We prove liminfn→∞g3(n)/3n=0. More precisely, for every ε>0 there is an integer d≥2 such that g3(dℓ)≤ ε3dℓ for all sufficiently large ℓ. The proof uses Korsky's characterization of the problem in terms of ternary coefficient sums and a consequence of an OpenAI construction that provides positive integer coefficients whose linear form is injective on large integer boxes. A base-three expansion then gives the result.
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BibTeX
@misc{costa2026negative,
title = {{A negative answer to the Erdős-Sárkőzy question}},
author = {Simone Costa},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06303},
eprint = {2609.06303},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06303},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A negative answer to the Erdős-Sárkőzy question
AU - Simone Costa
PY - 2026
DA - 2026/09/05
PB - arXiv
DO - 10.48550/arxiv.2609.06303
UR - https://arxiv.org/abs/2609.06303
AB - For a finite set A of positive integers, let H(A) be its set of subset sums, and let g_3(n) be the least N for which some n-element set A⊆{1,…,N} has H(A) free of nonconstant three-term arithmetic progressions. Erdős and Sárkőzy asked whether g_3(n)≫ 3^n. We prove liminf_n→∞g_3(n)/3^n=0. More precisely, for every ε>0 there is an integer d≥2 such that g_3(dℓ)≤ ε3^dℓ for all sufficiently large ℓ. The proof uses Korsky's characterization of the problem in terms of ternary coefficient sums and a consequence of an OpenAI construction that provides positive integer coefficients whose linear form is injective on large integer boxes. A base-three expansion then gives the result.
ER -