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Claim. Theorem 1.1 of Nets Hawk Katz and Terence Tao, Bounds on arithmetic projections, and applications to the Kakeya conjecture, Math. Res. Lett. 6 (1999), no. 6, 625–630, digested on the card [[../library/additive_combinatorics/katz_1999_bounds_arithmetic_projections_applications_kakeya_conjecture/_index|Katz and Tao 1999]]: if $A$, $B$, $C$ are finite subsets of an abelian group with at most $N$ elements each and $G\subseteq A\times B$ satisfies $a+b\in C$ for all $(a,b)\in G$, then $\#\{a-b:(a,b)\in G\}\le N^{2-1/6}$.

The bridge to Problem 1097 is not in the paper; it is stated here. Let $A$ be a set of $n$ integers, and take $B=A$, $C=2\cdot A=\{2a:a\in A\}$ and $G=\{(a,c)\in A^2:a+c\in2\cdot A\}$, so $N=n$. If $a,a+d,a+2d$ is a progression in $A$, then $(a+2d,a)\in G$ and its difference is $2d$; as $d\mapsto2d$ is injective, the number $D(A)$ of common differences is at most $\#\{a-c:(a,c)\in G\}\le n^{11/6}$.

The paper also gives a digit example in the integers with $\#\{a-b\}$ as large as $N^{\log6/\log3}$ under the theorem's hypotheses; through the embedding of [[problems/additive_combinatorics/E1097/claims/2014_04_14_lemm|Lemm's claim page]] it gives sets with about $n^{1.63}$ common differences, another negative answer to the second question, which Lemm's exponent supersedes.

Covers. The upper bound $D(A)\le n^{11/6}$ for the first question, the bound the site's commentary credits. 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 Mathematical Research Letters in 1999. The site's commentary credits the upper bound $11/6$ to this paper, 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 June 1999, which already states Theorem 1.1 with the exponent $2-1/6$ and the digit example.