Quantify the o(N) term in the best known upper bound for S(N)
Determine a reasonable explicit quantitative upper bound for the o(N) term in the inequality S(N) ≤ N/3 + o(N) obtained via the arithmetic regularity lemma, improving the current ineffective rate.
References
Their proof employs an elegant argument based on the arithmetic regularity lemma which leads to a more-or-less ineffective bound for $o(N)$; determining a reasonable upper bound remains an interesting problem.
— Large sum-free subsets of sets of integers via $L^1$-estimates for trigonometric series
(Bedert, 12 Feb 2025) in Section 1 (Introduction), discussion of upper bounds