Problem 153

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claims/: The 3 claim pages of Problem 153, one per claimant's result; the problem's standing derives from them.

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Statement. Let $A$ be a finite Sidon set and $A+A=\{s_1<\cdots<s_t\}$. Is it true that

$$\frac{1}{t}\sum_{1\leq i<t}(s_{i+1}-s_i)^2 \to \infty$$

as $\lvert A\rvert\to \infty$?

Status. Open, the site's label (OPEN). Three claim pages are recorded, two pending partial claims and one withdrawn full claim. Liu's withdrawn proof, a note of 2026-05-16 posted to the site's discussion thread, derived the answer yes from a shifted-intersection bound that was retracted the same day, and the author withdrew it. [[problems/additive_bases/E0153/claims/2026_05_19_liu|Liu's divergence for asymptotically maximum Sidon sets]], the corrected note dated 2026-05-20, entered in the author's repository on 2026-05-19 and posted to the thread on 2026-05-20, proves the answer yes for every family of Sidon sets whose diameter is $(1+o(1))n^2$ through Pikhurko's uniformity lemma. [[problems/additive_bases/E0153/claims/2026_08_14_kapoor|Kapoor's logarithmic lower bound]], a write-up of 2026-08-14, entered on the site's proof-claims thread on 2026-08-21, bounds the mean squared gap below by a constant times $\min\{\log\frac1{\kappa-1},\log n\}$ with $\kappa$ the diameter over $n^2$, answering yes for sets of nearly minimal diameter and for dyadically non-concentrated families and reducing the general case without settling it. The proof-claims thread had no comment on it as of 2026-10-06.

Source. erdosproblems.com/153, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #153, https://www.erdosproblems.com/153.

Formalization. Statement in formal-conjectures, tagged research open with no formal_proof attribute.

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