Year: 2026
Type: preprint
Abstract: Let $F$ be the set of positive cube-full integers. We prove that, for every $\varepsilon>0$, there is a reduced residue class in which $F+F+F$ has upper relative density at most $\varepsilon$. Consequently, the positive integers not representable as sums of at most three cube-full numbers have positive lower natural density. This proves the infinitude assertion in Erdős Problem \#940 for $r=3$. The argument combines a cubic-character estimate with a truncation of the parametrisation by the canonical 4-full part.
Source: arXiv (Cornell University)
Author: de Ryke, Basile Beyer
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214989629