PreprintCC BY 4.0arXiv
Affine Copies of Three-Point Patterns in Sets of Integers
Abstract
Let P={0,a,b}, where 0<a<b and gcd(a,b)=1. For a finite set A⊂ℤ, let MP+(A) count the copies x,x+ad,x+bd∈ A with d>0, and let MP(A) count the copies with any d≠0. We prove that every such three-point pattern other than the arithmetic progression {0,1,2} satisfies MP+(A)≤ 99/400|A|2+O(|A|), MP(A)≤ 13/28|A|2+OP(|A|). For the particular pattern P={0,1,3} -- the subject of a question raised by Ganguly and recorded as Problem 24 in Green's list of open problems -- we sharpen the bound allowing both signs of the dilation to M{0,1,3}(A)≤ 47/122|A|2+O(|A|).
Related
- Wild frieze patterns over the integers
- Cubic Optimization over Three Nonnegative Integers Is NP-hard
- Multi-Point Affine Bialgebras for AdS/CFT Integrability
- Modular properties of affine SLAsl2 torus n-point functions
- Algebraic and Symmetrized Affine star-Ricci Solitons for the Bott Connection on Three-Dimensional Lorentzian Lie Groups
BibTeX
@misc{korsky2026affine,
title = {{Affine Copies of Three-Point Patterns in Sets of Integers}},
author = {Samuel Korsky},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02308},
eprint = {2609.02308},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02308},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Affine Copies of Three-Point Patterns in Sets of Integers
AU - Samuel Korsky
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.02308
UR - https://arxiv.org/abs/2609.02308
AB - Let P={0,a,b}, where 0<a<b and gcd(a,b)=1. For a finite set A⊂ℤ, let M_P^+(A) count the copies x,x+ad,x+bd∈ A with d>0, and let M_P(A) count the copies with any d≠0. We prove that every such three-point pattern other than the arithmetic progression {0,1,2} satisfies M_P^+(A)≤ 99/400|A|^2+O(|A|), M_P(A)≤ 13/28|A|^2+O_P(|A|). For the particular pattern P={0,1,3} -- the subject of a question raised by Ganguly and recorded as Problem 24 in Green's list of open problems -- we sharpen the bound allowing both signs of the dilation to M_{0,1,3}(A)≤ 47/122|A|^2+O(|A|).
ER -