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Updated Oct 7

Problem 829

Statement. Let $A\subset\mathbb{N}$ be the set of cubes. Is it true that

1_A\ast 1_A(n) \ll (\log n)^{O(1)}?

Status. Open.

Source. erdosproblems.com/829, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #829, https://www.erdosproblems.com/829.

References.

[Ma35b] Mahler, Kurt, On the Lattice Points on Curves of Genus 1. Proc. London Math. Soc. (2) (1935), 431-466.
[St08] Stewart, Cameron L., Cubic Thue equations with many solutions. Int. Math. Res. Not. IMRN (2008), Art. ID rnn040, 11.

Formalization. Statement in formal-conjectures.

Progress

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Known Results

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Linked library material

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Linked from (8)

Diophantine Problems and Powers diophantine_problems/mahler_1935_lattice_points_curves_genus_1 Theorem 1 (p. 447): some k with 0 < |k| <= e^{gamma t^4} has at least t representations by the cubic form F Theorem 5 (p. 457): Theorem 1 with the t solutions confined to an angle about the origin Theorem 6 (p. 458): infinitely many k_v with more than (log k_v)^{1/4} representations as a sum of two positive cubes diophantine_problems/stewart_2008_cubic_thue_equations_many_solutions Theorem 1.1: a cubic Thue equation F(x, y) = m has at least c(log m)^(1/2) integer solutions for infinitely many m Theorem 4.1: many cube-free d up to T for which x^3 + axy^2 + by^3 = d is an elliptic curve of rank at least 2

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