Dense Sets with Prescribed Difference Representation Functions
Abstract
For a set A of positive integers, let dA(n) denote the number of representations of n as a difference of two elements of A. We prove that, if liminfn→∞f(n)≥2, then for every Sidon set B and every positive function ω(x) tending to infinity there is a set A⊆N such that dA(n)=f(n) for all n∈N and |A(x)−B(x/2)|≤ω(x) for all x≥1. We also give two extensions that allow f(n)=1 for infinitely many n. For an infinite Sidon set B, the same density scale B(x/2) is retained when f is at least $2$ on arbitrarily long intervals centered at elements of 2B. If {n:f(n)≥2} contains arbitrarily long intervals, the density scale B(x/3) is obtained for every Sidon set B. The proofs use remote insertion of new representations together with controlled deletions from a Sidon core. As a quantitative consequence, Ruzsa's dense infinite Sidon sequence gives A(x)=x√2−1+o(1) whenever {n:f(n)≥2} contains arbitrarily long intervals.
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BibTeX
@misc{ma2026dense,
title = {{Dense Sets with Prescribed Difference Representation Functions}},
author = {Wu-Xia Ma and Bing-Ling Wu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.27786},
eprint = {2610.27786},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.27786},
}
RIS
TY - UNPB
TI - Dense Sets with Prescribed Difference Representation Functions
AU - Wu-Xia Ma
AU - Bing-Ling Wu
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.27786
UR - https://arxiv.org/abs/2610.27786
AB - For a set A of positive integers, let d_A(n) denote the number of representations of n as a difference of two elements of A. We prove that, if liminf_n→∞f(n)≥2, then for every Sidon set B and every positive function ω(x) tending to infinity there is a set A⊆N such that d_A(n)=f(n) for all n∈N and |A(x)−B(x/2)|≤ω(x) for all x≥1. We also give two extensions that allow f(n)=1 for infinitely many n. For an infinite Sidon set B, the same density scale B(x/2) is retained when f is at least $2$ on arbitrarily long intervals centered at elements of 2B. If {n:f(n)≥2} contains arbitrarily long intervals, the density scale B(x/3) is obtained for every Sidon set B. The proofs use remote insertion of new representations together with controlled deletions from a Sidon core. As a quantitative consequence, Ruzsa's dense infinite Sidon sequence gives A(x)=x^√2−1+o(1) whenever {n:f(n)≥2} contains arbitrarily long intervals.
ER -