PreprintCC BY 4.0arXiv
A sharp lower bound for h-fold restricted sumsets of integers
Abstract
Let A⊆[0, L] be a set of kgeqslant 9 integers such that 0, L∈ A and gcd(A)=1. Define the h-fold restricted sumset h∧A={a1+⋯+ah: ai∈ A, 1leqslant ileqslant h, ai≠ aj for i≠ j}. In this paper, we prove that, for every 1leqslant hleqslant k−1, we have |h∧A|geqslant h(k−h)−k+2+min{L, h(k−h)−1h}, where 1h=1 for h∈{2, k−2} and 1h=0 otherwise. The bound is sharp for every 1leqslant hleqslant k−1 and the cardinality threshold kgeqslant 9 cannot be decreased. We also give a lower bound for the union of two consecutive restricted sumsets in a cyclic group and characterize equality.
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BibTeX
@misc{yujie2026sharp,
title = {{A sharp lower bound for $h$-fold restricted sumsets of integers}},
author = {Yujie and Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.25073},
eprint = {2610.25073},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.25073},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A sharp lower bound for h-fold restricted sumsets of integers
AU - Yujie
AU - Wang
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2610.25073
UR - https://arxiv.org/abs/2610.25073
AB - Let A⊆[0, L] be a set of kgeqslant 9 integers such that 0, L∈ A and gcd(A)=1. Define the h-fold restricted sumset h^∧A={a_1+⋯+a_h: a_i∈ A, 1leqslant ileqslant h, a_i≠ a_j for i≠ j}. In this paper, we prove that, for every 1leqslant hleqslant k−1, we have |h^∧A|geqslant h(k−h)−k+2+min{L, h(k−h)−1_h}, where 1_h=1 for h∈{2, k−2} and 1_h=0 otherwise. The bound is sharp for every 1leqslant hleqslant k−1 and the cardinality threshold kgeqslant 9 cannot be decreased. We also give a lower bound for the union of two consecutive restricted sumsets in a cyclic group and characterize equality.
ER -