Erdős #1206 · number theory / sidon sets

This question may also be asked for higher powers, and [324] suggests that for fifth powers $\{n^5 :n\geq 1\}$ may itself be a Sidon set.

Gabdullin and Konyagin [GaKo24] proved that there is a constant $c>0$ such that $\{ n^3 : N-cN^{1/2}\leq n\leq N\}$ is a Sidon set. Garaev, Garayev, and Konyagin [GGK26] have proved that the exponent $1/2$ can be improved to $4/7-o(1)$ for infinitely many $N$, and to $3/5$ for all $N$ if we replace $n^3$ with $n^4$.

In [Er80] Erdős defines $g_k(A)$ to be the maximal size of a Sidon set of $\{a^k : a\in A\}$, and asks whether $g_k(A)\geq g_k(\{1,\ldots,N\})$ where $N=\lvert A\rvert$. (See [530] for the linear analogue.)

See also [773] for an analogous question about squares.

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