Skip to content
DraconiticResearch
Search Sources API
203,203 studies31 sourcesUpdated 14 min ago
PreprintCC BY-NC-SA 4.0arXiv

Multiple representations in difference sets under polynomial density constraints

  • Yuchen Ding
  • Sándor Z. Kiss
  • Wei Niu
  • Péter Pál Pach
  • Csaba Sándor
Open at sourcePDFDOICite

Abstract

Motivated by the classical study of additive representation functions and the combinatorial phenomenon of sum-difference asymmetry, we investigate the subtractive representation function dℬ,𝒜(n), which counts the number of solutions of the equation n = b − a with a∈ 𝒜 and b∈ ℬ. Specifically, we study lower bounds and the asymptotic behavior of dℬ,𝒜(n) under polynomial density constraints of orders xα and x1−α, where 0 < α< 1. We first establish a general lower bound for the first moment of dℬ,𝒜(n). Furthermore, by employing a recursive construction, we prove that this lower bound is essentially optimal under certain regularity assumptions on the counting functions. Finally, utilizing probabilistic methods, we show the existence of sets for which dℬ,𝒜(n) has a logarithmic point-wise upper bound.

Related

  1. Polynomial Difference Schemes with Energy and Momentum Multipliers on Orthogonal Uniform Meshes: Stencil Restrictions and Monge--Ampère-Type Examples
    E. I. Kaptsov
    arXivPreprintarXiv licenceFull textMathematical physicsNumerical analysis & scientific computingarXiv
  2. Open-Loop Stackelberg LQ Difference Games with Coupled-Affine Inequality Constraints: An Exact OCP--LCS/LCQP Reformulation
    Shrestha Ghosh, Puduru Viswanadha Reddy
    arXivPreprintarXiv licenceFull textOptimisation & controlarXiv
  3. DAGR: State-Conditioned Goal Representations via Difference-Aware Goal Cross-Attention
    Xing Lei, Wenyan Yang, Xuetao Zhang +1
    arXivPreprintarXiv licenceFull textStatisticsarXiv
  4. A Monte Carlo Estimator for an Isolated Polynomial Zero via Contour Integral Representations
    Athanasios Christou Micheas
    arXivPreprintarXiv licenceFull textAnalysis & PDEsStatisticsarXiv
  5. Linear Programming Representations and Strongly Polynomial Algorithms for Robust Markov Decision Processes
    Han Zhong, Yinyu Ye
    arXivPreprintarXiv licenceFull textOptimisation & controlarXiv

Cite

BibTeXRIS
BibTeX
@misc{ding2026multiple,
  title = {{Multiple representations in difference sets under polynomial density constraints}},
  author = {Yuchen Ding and Sándor Z. Kiss and Wei Niu and Péter Pál Pach and Csaba Sándor},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.40489},
  eprint = {2610.40489},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.40489},
  note = {CC BY-NC-SA 4.0},
}
RIS
TY  - UNPB
TI  - Multiple representations in difference sets under polynomial density constraints
AU  - Yuchen Ding
AU  - Sándor Z. Kiss
AU  - Wei Niu
AU  - Péter Pál Pach
AU  - Csaba Sándor
PY  - 2026
DA  - 2026/10/29
PB  - arXiv
DO  - 10.48550/arxiv.2610.40489
UR  - https://arxiv.org/abs/2610.40489
AB  - Motivated by the classical study of additive representation functions and the combinatorial phenomenon of sum-difference asymmetry, we investigate the subtractive representation function d_ℬ,𝒜(n), which counts the number of solutions of the equation n = b − a with a∈ 𝒜 and b∈ ℬ. Specifically, we study lower bounds and the asymptotic behavior of d_ℬ,𝒜(n) under polynomial density constraints of orders x^α and x^1−α, where 0 < α< 1. We first establish a general lower bound for the first moment of d_ℬ,𝒜(n). Furthermore, by employing a recursive construction, we prove that this lower bound is essentially optimal under certain regularity assumptions on the counting functions. Finally, utilizing probabilistic methods, we show the existence of sets for which d_ℬ,𝒜(n) has a logarithmic point-wise upper bound.
ER  -