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Claim. For a finite Sidon set $A$ write $n=\lvert A\rvert$, $\kappa=\operatorname{diam}(A)/n^2$, $t=\lvert A+A\rvert$ and

$$Q(A)=\frac1t\sum_{1\le i<t}(s_{i+1}-s_i)^2,\qquad A+A=\{s_1<\cdots<s_t\},$$

the quantity of Problem 153. The write-up states that there is a constant $c>0$ with

$$Q(A)\ge c\min\Bigl\{\log\frac1{\kappa-1},\;\log n\Bigr\}$$

for every Sidon set of large enough size. The proof decomposes by scale: the interval spanned by $A$ is split into dyadic pieces, Cauchy–Schwarz is applied within each piece, and an energy inequality for the piece densities, $\sum_k m_k^2/\ell_k\le1+o(1)$ in the author's notation, forces those densities down; the contributions of the scales add up to the logarithm. The author names Claude Opus 5, run at xhigh effort, as the system used in the work; the write-up entered the author's repository on 2026-08-14 (the preprint link is pinned to that commit) and was entered on the site's proof-claims thread on 2026-08-21.

Submission note. Posted to erdosproblems.com as a proof claim by Rajveer Kapoor (account RajveerKapoor) on 21 August 2026, giving "Claude Opus 5 (xhigh effort)" as the AI used:

Write $n=|A|$, $\kappa=\operatorname{diam}(A)/n^2$, $t=|A+A|$, $Q(A)=t^{-1}\sum(s_{i+1}-s_i)^2$. Theorem. There is $c>0$ with $Q(A)\ge c\min{\log\frac1{\kappa-1},\log n}$ for all large $n$. Proof by scale decomposition: cut the interval dyadically, apply Cauchy-Schwarz per window, and use the sharp energy inequality $\sum_k m_k^2/\ell_k\le1+o(1)$ to force the densities down. Summing scales gives the logarithm. So the answer is yes for every family with $\operatorname{diam}(A_n)=(1+o(1))n^2$ (Singer, Bose-Chowla, Ruzsa, affine images), and for every dyadically non-concentrated family. Also $\liminf f(n)\ge16$, improved to $\approx16.13$. Reduction: the answer is yes unless there exist $C\ge1$, $\alpha<1$ and Sidon $A_n$ with $\operatorname{diam}A_n\le Cn^2$ and $|A_n\cap[0,N2^{-j}]|\ge C^{-1}n2^{-\alpha j}$ at both ends for all $j\le\log_2n$. Such a family needs $\kappa\in[1+e^{-O(C)},\sqrt C/4]$ and $\alpha>1/2$. Not claimed: the general case. Notes: Partial result; the status of #153 should not change. leon2k2k2k (20 May 2026) already claimed the asymptotically maximum case via Pikhurko uniformity. The theorem recovers that as the $\kappa\to1$ endpoint; what is new is that it survives $\kappa$ bounded away from 1, plus the reduction. One bottleneck, offered as data and not as an obstruction: the relaxation generated by interval Cauchy-Schwarz, the prefix bound, the energy inequality and the local Sidon bound has a finite optimum, with witness $p_j=\frac12 2^{-0.75(j-1)}$, $\kappa=1.1222$, cost $6.83$. That is where my own attempts stalled. I would not read it as showing pair-counting cannot settle the problem: a different relaxation, an extra constraint or a sharper form of any of the four could well get past it. Code and the LP witness: https://github.com/RajveerKapoor/erdos-work

Covers. The answer yes for every family of Sidon sets whose diameter is $(1+o(1))n^2$, since there $\kappa\to1$ and the bound tends to infinity; by the write-up's Corollary 7 this includes the Singer, Bose–Chowla and Ruzsa sets, their affine images and all $o(n)$-deletions. By its Corollary 8 the answer is also yes for every family the author calls dyadically non-concentrated, whatever $\kappa$ is, which covers the Erdős–Turán sets $\{2pk+(k^2\bmod p)\}$, for which $\kappa=2$. Also claimed: with $f(n)$ the least value of $Q(A)$ over Sidon sets of size $n$, $\liminf f(n)\ge16$, improved to about $16.13$. The general case is reduced, not settled: by the write-up the answer is yes unless there are $C\ge1$, $\alpha<1$ and Sidon sets $A_n$ with $\operatorname{diam}A_n\le Cn^2$ that keep at least $C^{-1}n2^{-\alpha j}$ elements in the initial and final segments of length $2^{-j}$ times the diameter for every $j\le\log_2n$; such a family needs $\kappa\in[1+e^{-O(C)},\sqrt C/4]$ and $\alpha>1/2$. The author does not claim the general case, and the site's label is unchanged.

Standing. Claimed. The thread entry had no comments as of 2026-10-06 and the site's label is OPEN; no named mathematician has examined the write-up, there is no refereed publication and no Lean development. The author's thread note credits the forum post of 2026-05-20 that proved the asymptotically maximum case through Pikhurko's uniformity lemma, [[problems/additive_bases/E0153/claims/2026_05_19_liu|Liu's divergence for asymptotically maximum Sidon sets]], which the theorem above recovers as its $\kappa\to1$ endpoint. The write-up states as a barrier (its Theorem 9) that the relaxation built from the window Cauchy–Schwarz step, a prefix bound, the energy inequality and the local Sidon bound has a finite optimum, with an explicit witness profile at $\kappa=1.1222$, so that no combination of the pair-counting inequalities it uses can prove $Q\to\infty$; the author's thread note offers the same result as data rather than as an obstruction, leaving open that a different relaxation might pass.

Depends on. Nothing in this wiki.