Maximising the number of solutions to linear equations

Abstract

We study the asymptotically maximal possible number of integer solutions to the linear equation ax+by+cz = 0 with a fixed choice of a, b, c ∈ ℤ and variables x, y, z ∈ S for some finite set S⊂ ℤ, as |S|→ +∞. Define γ_{a, b, c} to be the largest constant for which there are arbitrary large finite sets S⊂ ℤ such that the number of solutions to ax+by+cz=0 with x,y,z∈ S is γ_a,b,c|S|²−o(|S|²). We prove structural results for general $a, b, c$ and moreover, we show that 5/13≤ γ_1,1,−3≤ 1/2−δ for some constant δ>0. In addition we show that the limit as a → ∞ of γ_1,1,−a is equal to precisely 1/5.

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BibTeX RIS

BibTeX

@misc{ranđelovic2026maximising,
  title = {{Maximising the number of solutions to linear equations}},
  author = {Žarko Ranđelović and Xuancheng Shao and Max Wenqiang Xu and Shengtong Zhang and Yuan Zhou},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.06975},
  eprint = {2609.06975},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.06975},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Maximising the number of solutions to linear equations
AU  - Žarko Ranđelović
AU  - Xuancheng Shao
AU  - Max Wenqiang Xu
AU  - Shengtong Zhang
AU  - Yuan Zhou
PY  - 2026
DA  - 2026/09/07
PB  - arXiv
DO  - 10.48550/arxiv.2609.06975
UR  - https://arxiv.org/abs/2609.06975
AB  - We study the asymptotically maximal possible number of integer solutions to the linear equation ax+by+cz = 0 with a fixed choice of a, b, c ∈ ℤ and variables x, y, z ∈ S for some finite set S⊂ ℤ, as |S|→ +∞. Define γ_a, b, c to be the largest constant for which there are arbitrary large finite sets S⊂ ℤ such that the number of solutions to ax+by+cz=0 with x,y,z∈ S is γ_a,b,c|S|^2−o(|S|^2). We prove structural results for general $a, b, c$ and moreover, we show that 5/13≤ γ_1,1,−3≤ 1/2−δ for some constant δ>0. In addition we show that the limit as a → ∞ of γ_1,1,−a is equal to precisely 1/5.
ER  -

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