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Claim. For every sequence of finite Sidon sets $A_n$ with $\lvert A_n\rvert\to\infty$, the mean squared gap $Q(A_n)$ of Problem 153 tends to infinity; the answer yes to the whole question. The note Erdős #153: Finite mean-square gap divergence by Yu Leon Liu, dated 2026-05-16, entered the author's repository and was posted to the site's discussion thread that day (the preprint link is pinned to the commit that added it). Its Theorem 3 derives the claim from a single lemma, the bound $\lvert(A+A)\cap(A+A+d)\rvert\le2\lvert A\rvert^{3/2}$ for every nonzero shift $d$, which the note attributes to the proof of a theorem of Erdős, Sárközy and Sós and to a comment on the thread of Problem 152: a gap of length $d$ between consecutive sums yields a distinct element of the shifted intersection, so few gaps are short and the squared gaps add up to a large total. The author's post says that the proof was found with the help of GPT-5.5 and Rethlas and verified by hand.

Withdrawal. The bound the proof rests on was retracted on the thread of Problem 152 by the site's curator, T. F. Bloom, on 2026-05-16, hours after the note was posted. The author marked the proof incorrect, pulled the note on 2026-05-19 and replaced it with a corrected note that proves only the asymptotically maximum case, [[problems/additive_bases/E0153/claims/2026_05_19_liu|Liu's divergence for asymptotically maximum Sidon sets]]. The problem's standing takes nothing from this page.

Depends on. Nothing in this wiki.