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Claim. Theorem 2.1 of Marius Lemm, New counterexamples for sums-differences, Proc. Amer. Math. Soc. 143 (2015), no. 9, 3863–3868, digested on the card [[../library/additive_combinatorics/lemm_2015_new_counterexamples_sums_differences/_index|Lemm 2015]], states that there is an $\alpha>1.77898$ for which the statement $\mathrm{SD}(0,1,\infty;\alpha)$ fails. In the paper's notation, with $\pi_r(a,b)=a+rb$ and $\pi_\infty(a,b)=b$, this says that there are finite sets $G$ of pairs on which $a-b$ is injective and $\log\lvert G\rvert\ge\alpha\max(\log\lvert A\rvert,\log\lvert B\rvert,\log\lvert A\overset{G}{+}B\rvert)$, where $A$ and $B$ are the sets of first and second coordinates of $G$ and $A\overset{G}{+}B=\{a+b:(a,b)\in G\}$; the construction gives such sets with $\max(\lvert A\rvert,\lvert B\rvert,\lvert A\overset{G}{+}B\rvert)=N$ arbitrarily large and $\lvert A\overset{G}{-}B\rvert=\lvert G\rvert\ge N^{\alpha-\varepsilon}$ for every $\varepsilon>0$. The sets are built from integer lattice points, and a linear map with widely spaced coefficients carries them into the integers while keeping every sum and difference distinct.

The bridge to Problem 1097 is not in the paper; it is stated here. Given such $G$, the set $X=2\cdot A\cup2\cdot B\cup(A\overset{G}{+}B)$ of integers has at most $3N$ elements, and each $(a,b)\in G$ with $a\ne b$ gives the progression $2a,a+b,2b$ in $X$ with common difference $b-a$. Hence $D(X)\ge\lvert A\overset{G}{-}B\rvert-1\ge N^{\alpha-\varepsilon}-1$, and since $\alpha>1.77898$ there are sets $X$ of arbitrarily large size $n$ with more than $n^{1.77898}$ common differences. The embedding is the observation of Koishi Chan on the problem's discussion thread of 2 December 2025, which the site's commentary adopts. The improvement by AlphaEvolve [GGTW25] raises Lemm's exponent only in the eighth decimal.

Covers. The second question: $O(n^{3/2})$ common differences do not always suffice. The first question, the order of magnitude, is not settled.

Depends on. No page of this wiki: the embedding is proved above.

Acceptance. Refereed: the paper appeared in Proceedings of the American Mathematical Society in 2015. The site's commentary credits the lower bound to this paper and says it answers the second question negatively, but the site labels the problem OPEN, so the commentary is not acceptance and no reviewed is listed. The page is dated by the arXiv preprint of 14 April 2014.