PreprintCC BY 4.0arXiv
Subset sums of power values over Galois rings
Abstract
Let R=GR(pm,pmf), where p is prime, q=pf, m,f≥1, d≥2 and pnmid d. We count n-element subsets of P={xd:x∈ R}∖{0}, the set of distinct nonzero dth-power values, with a prescribed sum. For fixed m,d,n with n≥3, the count equals q−mbinom|P|n(1+Om,d,n(q1−n/2)) as q→∞ through admissible prime powers, uniformly over all targets and with both p and f allowed to vary. For fixed m,d, we also classify uniform relative equidistribution at the remaining positive sizes: it fails at n=1, and at n=2 it holds along an admissible sequence if and only if, for all sufficiently large terms, gcd(d,q−1)=1 and (m≥2 or p≠2).
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BibTeX
@misc{gu2026subset,
title = {{Subset sums of power values over Galois rings}},
author = {Zijie Gu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.28837},
eprint = {2610.28837},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.28837},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Subset sums of power values over Galois rings
AU - Zijie Gu
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.28837
UR - https://arxiv.org/abs/2610.28837
AB - Let R=GR(p^m,p^mf), where p is prime, q=p^f, m,f≥1, d≥2 and pnmid d. We count n-element subsets of P={x^d:x∈ R}∖{0}, the set of distinct nonzero dth-power values, with a prescribed sum. For fixed m,d,n with n≥3, the count equals q^−mbinom|P|n(1+O_m,d,n(q^1−n/2)) as q→∞ through admissible prime powers, uniformly over all targets and with both p and f allowed to vary. For fixed m,d, we also classify uniform relative equidistribution at the remaining positive sizes: it fails at n=1, and at n=2 it holds along an admissible sequence if and only if, for all sufficiently large terms, gcd(d,q−1)=1 and (m≥2 or p≠2).
ER -