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Updated Oct 7

Problem 1206

Statement. Does $\{1,2^3,\ldots,N^3\}$ contain a Sidon set of size $\gg N$ ?

Is there an infinite set $A\subset \mathbb{N}$ of positive density such that $\{a^3 : a\in A\}$ is a Sidon set?

Status. Open.

Source. erdosproblems.com/1206, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1206, https://www.erdosproblems.com/1206.

References.

[Er80] Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115.
[GGK26] M. Garaev, F. Garayev, and S. Konyagin, On Sidon sets with squares, cubes, and quartics in short intervals. arXiv:2602.08807 (2026).
[GaKo24] Gabdullin, M. R. and Konyagin, S. V., Trigonometric polynomials with frequencies in the set of cubes. Math. Notes (2024), 336-340.

Formalization. Statement in formal-conjectures.

Progress

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Known Results

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Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

Linked from (8)

Additive Bases and Sidon Sets additive_bases/gabdullin_2024_trigonometric_polynomials_frequencies_set_cubes Theorem 1.1: cubes n^3 with N <= n <= N + N^(2/3-eps) satisfy ||f||_4 << eps^(-1/4) ||f||_2 Theorem 1.2: the cubes n^3 with N <= n <= N + (0.5N)^(1/2) form a Sidon set, sharp up to the constant additive_bases/garaev_2026_sidon_sets_squares_cubes_quartics_short Theorem 3 (p. 3): the cubes n^3 with N <= n < N + (38N/3 + 1297/36)^(1/2) + 19/6 form a Sidon set Theorem 4 (p. 3): the cubes n^3 with N <= n <= N + cN^(2/3) are never a Sidon set Theorem 5 (p. 4): for infinitely many N the cubes n^3 with N <= n <= N + N^(4/7-eps) form a Sidon set

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