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I posted this initially on SE, but after I didn't found a particular reference on it, I decided it would be more appropriate to post it here. A friend shared this observation with me and I thought ...

Fix $\epsilon>0$. For all large $N$, does there exist $A\subset [N]:=\{1,\dots,N\}$ such that both $A+A$ and $A^c:=[N]\setminus A$ lack arithmetic progressions of length $N^\epsilon$? I am aware ...

Let $X$ be an infinite sequence of integers$$x_1 < x_2 < x_3 < \ldots,$$and let $X(n)$ be the set$$\{x_1, x_2, \ldots, x_n\}.$$ Question. What is known about $X$ when we have$$|X(n) + X(n)| &...