PreprintarXiv licencearXiv
A density deficit for sums of three cube-full numbers
Abstract
Let F be the set of positive cube-full integers. We prove that, for every ε>0, there is a reduced residue class in which F+F+F has upper relative density at most ε. 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.
Related
- Dynamics of the Density Cube
- Tree-indexed sums of Catalan numbers
- Relative isoperimetry in the three-dimensional cube: classification and quantitative stability
- Fixed-Radius Neighbourhoods in Three-Dimensional Unit-Cube Graphs
- Codimension-three regularity of noncollapsed Ricci limit spaces under an integral volume-deficit bound
BibTeX
@misc{ryke2026density,
title = {{A density deficit for sums of three cube-full numbers}},
author = {Basile Beyer de Ryke},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.35772},
eprint = {2609.35772},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.35772},
}
RIS
TY - UNPB
TI - A density deficit for sums of three cube-full numbers
AU - Basile Beyer de Ryke
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.35772
UR - https://arxiv.org/abs/2609.35772
AB - Let F be the set of positive cube-full integers. We prove that, for every ε>0, there is a reduced residue class in which F+F+F has upper relative density at most ε. 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.
ER -