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Write $S_r(N)$ for the largest size of a set $A\subseteq\{1,\ldots,N\}$ in which every integer has at most $r$ representations $a+a'$ with $a\le a'$, and $D_r(N)$ for the largest size of a set $B\subseteq\{1,\ldots,N\}$ in which every nonzero difference has at most $r$ ordered representations $b-b'$. Ho proves (Theorem 1 of the write-up) that for every fixed $r\geq 2$, with $s=\lfloor r/2\rfloor$,

$$\limsup_{N\to\infty}\frac{D_r(N)}{\sqrt N} \leq \sqrt r < \frac{r+s}{\sqrt{r+2s}} \leq \liminf_{N\to\infty}\frac{S_r(N)}{\sqrt N}.$$

If the constants $c_r$ and $c_r'$ of the statement exist, then $c_r'\leq\sqrt r<c_r$, so both questions are answered yes: $c_r\neq c_r'$ and $c_r'<c_r$ for every $r\geq 2$. The separation holds without assuming that either limit exists.

The upper bound adapts the Erdős–Turán argument for Sidon sets: counting pairs of elements of $B$ inside windows of length $H$ by Cauchy–Schwarz and by the difference bound gives $|B|^2\leq rN+o(N)$ for $H=\lfloor N^{2/3}\rfloor$. The lower bound is the finite $B_2[g]$ construction of Cilleruelo, Ruzsa and Trujillo (Theorem 2.1 of cilleruelo_2002_upper_lower_bounds_finite_b_h, J. Number Theory 97 (2002), 26–34), re-proved in the write-up: a Singer cyclic Sidon set of size $q+1$ modulo $q^2+q+1$ is lifted along the pattern $\{0,\ldots,r-1\}\cup\{r+s,\ldots,r+2s-1\}$, which has at most $r$ ordered representations of every sum, and the prime number theorem supplies a prime $q$ of the right size. For $r=2$ the bounds read $c_2'\leq\sqrt2$ and $c_2\geq 3/2$.

The write-up was posted on 2026-04-22. A revision of 2026-05-03 adds an acknowledgement that GPT-5.4 Pro contributed materially to the arguments, proofs and exposition, and that the author accepts responsibility for the final text; the mathematics is unchanged. The site credits the observation to Ho and GPT-5.4 Pro.

Acceptance. The site's curator, T. F. Bloom, accepted the argument: the problem is labeled proved and the remarks state the two bounds and the resulting inequality (page last edited 2026-04-24), two days after the write-up was posted to the forum. That is the reviewed evidence. The construction half is refereed as part of the Cilleruelo–Ruzsa–Trujillo paper, but the write-up itself and the inequality it draws are not refereed, so the claim lists no refereed evidence.

Depends on. Nothing in this wiki; the inputs are the refereed construction cited above and the elementary window count.