Gaps in dilates of Bose sets, moments of dilated Singer sets, and the excess in Erdős's Sidon problem: numerical evidence

Abstract

Let h(N) be the maximal size of a Sidon set in {1,...,N}. Erdős asked whether h(N) = N^{1/2} + O(1) (Guy C9); the weaker question h(N) = N^{1/2} + O(N^ε) is Erdős problem #30, and the evidence here concerns the O(1) question only. Following a dilation idea, we study the largest cyclic gap over all dilates of Bose sets: a log q law fitted on primes q ≤ 1000 fits held-out primes up to 2003 much better than a constant and, frozen before computing, predicted 48 new primes 2011 ≤ q ≤ 5881 (RMSE 1.52 against 7.34). Random sets show the same growth. For Singer perfect difference sets we study factorial moments of the number of points of a random dilate in a window of length Cq: the second moment equals the hypergeometric one exactly, the third-moment deviation has the main term -3/(2Cq) coming from the three forbidden ratios {-1, 2, 1/2}, and the RMS deviation of ratio counts of height 3-6 is consistent with decay like q^{-1/2} up to q ≈ 3000. All results are numerical; no proof is claimed. Computations were designed and run with AI assistance (Anthropic Claude). Data and code: https://github.com/vichienFF/ramanujan-explorer/tree/main/erdos_sidon