Skip to content
DraconiticResearch
Search Sources API
193,349 studies31 sourcesUpdated 15 min ago
PreprintCC BY 4.0arXiv

Error Term Estimates for Sets of Remainders

  • Omkar Baraskar
  • Prashant Gokhale
  • Sarvagya Jain
  • Adam Kieżun
Open at sourcePDFDOICite

Abstract

For a positive integer n, let S1(n)={nbmod k:1≤ k≤⌊ n/2⌋}, s1(n)=|S1(n)|. Baraskar and Vukusic proved s1(n)=Ξn+O(n/(log nloglog n)) for an explicit positive constant Ξ, and asked for a sharper error term. Their computations suggested that an O(n1/3) bound might hold. We prove that, for every α>√2, limsupn→∞ (s1(n)−Ξn)/(nexp{−α√(log nloglog n)}) =∞, and the corresponding limit inferior is −∞. In particular, no fixed power saving is possible. We also prove the complementary upper estimate s1(n)=Ξn+O( nexp{−α√(log nloglog n)} ) for every fixed α∈ (0,√2), sharpening the previous O(n/(log nloglog n)) bound. Thus, the constant √2 is the threshold for bounds of the form nexp{−α√(log nloglog n)}.

Related

  1. A logarithmic error term for subset sums of zero-sum-free sets
    Chao Liu
    arXivPreprintarXiv licenceFull textAlgebra & number theoryCombinatorics & discrete mathematicsarXiv
  2. Asymptotic Behavior of Iterated Sets of Remainders
    Omkar Baraskar, Prashant Gokhale, Sarvagya Jain +1
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv
  3. Estimates on Fully Nonlinear Elliptic Equations with Unrectifiable Exceptional Gradient Sets
    Tuoc Phan
    arXivPreprintCC BY 4.0Full textAnalysis & PDEsGeometry & topologyarXiv
  4. A Priori and A Posteriori Error Estimates for a Crank Nicolson Type Scheme of an Elliptic Problem with Dynamical Boundary Conditions
    Rola Ali Ahmad, Toufic El Arwadi, Houssam Chrayteh +1
    arXivPreprintarXiv licenceFull textNumerical analysis & scientific computingarXiv
  5. Anisotropic error estimates for the L2-projection onto discontinuous finite element spaces
    Joaquín Valenzuela Díaz, Yuan Wang, Sander Rhebergen
    arXivPreprintCC BY 4.0Full textNumerical analysis & scientific computingarXiv

Cite

BibTeXRIS
BibTeX
@misc{baraskar2026error,
  title = {{Error Term Estimates for Sets of Remainders}},
  author = {Omkar Baraskar and Prashant Gokhale and Sarvagya Jain and Adam Kieżun},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.04993},
  eprint = {2610.04993},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.04993},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Error Term Estimates for Sets of Remainders
AU  - Omkar Baraskar
AU  - Prashant Gokhale
AU  - Sarvagya Jain
AU  - Adam Kieżun
PY  - 2026
DA  - 2026/10/04
PB  - arXiv
DO  - 10.48550/arxiv.2610.04993
UR  - https://arxiv.org/abs/2610.04993
AB  - For a positive integer n, let S_1(n)={nbmod k:1≤ k≤⌊ n/2⌋}, s_1(n)=|S_1(n)|. Baraskar and Vukusic proved s_1(n)=Ξn+O(n/(log nloglog n)) for an explicit positive constant Ξ, and asked for a sharper error term. Their computations suggested that an O(n^1/3) bound might hold. We prove that, for every α>√2, limsup_n→∞ (s_1(n)−Ξn)/(nexp{−α√(log nloglog n)}) =∞, and the corresponding limit inferior is −∞. In particular, no fixed power saving is possible. We also prove the complementary upper estimate s_1(n)=Ξn+O( nexp{−α√(log nloglog n)} ) for every fixed α∈ (0,√2), sharpening the previous O(n/(log nloglog n)) bound. Thus, the constant √2 is the threshold for bounds of the form nexp{−α√(log nloglog n)}.
ER  -