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Claim. Let $(A_n)_{n\ge1}$ be a sequence of Sidon sets with $\lvert A_n\rvert=n$ and $\operatorname{diam}(A_n)=(1+o(1))n^2$, the sets the note calls asymptotically maximum. Then, with $Q$ the quantity of Problem 153,

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

This is Theorem 2 of the note Erdős #153: Asymptotically maximum case by Yu Leon Liu, dated 2026-05-20, which entered the author's repository on 2026-05-19 (the preprint link is pinned to the revision linked from the thread) and was posted to the site's discussion thread on 2026-05-20. The proof has two inputs. Pikhurko's uniformity lemma, through the convolution argument of the author's note on Problem 819, gives the density of $A+A$ in a window of length $\eta D$ centered at $xD$, $D$ the diameter, as $\varphi(x/2)+O(\eta)+o_n(1)$ with $\varphi$ the triangular density, so the sumset thins out linearly at both ends of $[0,2D]$ (Lemma 4); a localization inequality bounds $\sum_i(s_{i+1}-s_i)^2$ below by $\sum_k\lvert I_k\rvert^2/(\lvert T\cap I_k\rvert+1)$ over disjoint windows $I_k$ (Lemma 5). Windows near the ends of the sumset then contribute a Riemann sum for $\int_\varepsilon^{1/2}du/u$, which grows without bound as $\varepsilon\to0$. The author's thread post says that the result was found with the help of GPT-5.5 and Rethlas and verified by hand; the note itself names no system.

Covers. Every family of Sidon sets with $\operatorname{diam}(A_n)=(1+o(1))n^2$, which by the note's footnote includes the Bose–Chowla sets. The note's Observation 1 records, after a thread comment of 2026-02-15, that Cauchy–Schwarz already gives $Q(A)\to\infty$ when $\operatorname{diam}(A)/n^2\to\infty$, so the open range is $\operatorname{diam}(A_n)=O(n^2)$; its Remark 3 leaves the case $\liminf\operatorname{diam}(A_n)/n^2>1$ open, and the note does not claim the general case. [[problems/additive_bases/E0153/claims/2026_08_14_kapoor|Kapoor's logarithmic lower bound]] of August 2026 recovers this result as the $\kappa\to1$ endpoint of its theorem, and its thread note credits this post.

Standing. Claimed. The note replaced the author's withdrawn full claim of 2026-05-16 ([[problems/additive_bases/E0153/claims/2026_05_16_liu|Liu's withdrawn proof]]), whose argument rested on a retracted bound; the corrected note uses neither that bound nor any result of the thread. As of 2026-10-06 the result is not on the site's proof-claims tab, the site's label is OPEN, and no comment on the post, refereed publication, outside review or Lean development is recorded.

Depends on. Nothing in this wiki.