On the diameter and distribution of Sidon sets

@misc{carter2026diameter,
  title = {{On the diameter and distribution of Sidon sets}},
  author = {Daniel Carter and Kevin O'Bryant and Terence Tao},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.36207},
  eprint = {2610.36207},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.36207},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - On the diameter and distribution of Sidon sets
AU  - Daniel Carter
AU  - Kevin O'Bryant
AU  - Terence Tao
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2610.36207
UR  - https://arxiv.org/abs/2610.36207
AB  - A Sidon set is a set 𝒜 of integers with the property that if 0≠ a−b=c−d and a,b,c,d∈𝒜, then (a,b)=(c,d). Our main result is an improved and explicit bound on the diameter of a Sidon set: if 𝒜 is a Sidon set with k≥ 1 elements, then diam(𝒜) ≥ k^2 (1/(1 + 0.945/√k)−0.945/√k) −1; in particular, diam(𝒜)≥ k^2−1.89 k^3/2. This implies that if 𝒜⊆[1,N] is a Sidon set, then |𝒜| ≤ N^1/2+0.945 N^1/4+0.447. Our technique also gives an improved bound for the case of large k, namely diam(𝒜) ≥ k^2−1.88607 k^3/2 for k≥ 483. A γ-Golomb ruler is a set 𝒜 of integers with the property that there are not γ+1 distinct pairs (a_i,b_i)∈𝒜^2 with 0≠ a_1−b_1=⋯=a_γ+1−b_γ+1. Our diameter bounds extend in a natural way to the case of Golomb rulers. Our techniques can also be adapted to address a number of questions on the distribution of Golomb rulers: we provide upper bounds on the two quantities |a_ℓ−a_1 −(ℓ−1)/(k−1)(a_k−a_1)| and max_1≤ i<k (a_i+1−a_i), where 𝒜={a_1<…<a_k} is a γ-Golomb ruler with diam(𝒜)≤ (k^2−bk^3/2)/γ (where b∈ ℝ is a free parameter).
ER  -