PreprintCC BY 4.0arXiv
Gaussian Lower Bounds for Distinct Subset Sums in Higher Dimensions
Abstract
For sequences of n nonnegative integer vectors in ℤk with distinct subset sums, we prove, for fixed k, the uniform lower bound M≥ (Ck−o(1))2n/k/√n for the largest coordinate, where Ck=√(2k/π) ((k+1)(k−1)/(2k))/(√(⌊ (k+1)2/4⌋)). The proof compares a Gaussian-kernel expectation with a sum over a translated lattice, uniformly over the vector sequence. We optimise the resulting quadratic-form constant using a vertex second-moment certificate. We show that Ck also exceeds the variance constant obtained by optimising an ellipsoidal geometry.
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BibTeX
@misc{gu2026gaussian,
title = {{Gaussian Lower Bounds for Distinct Subset Sums in Higher Dimensions}},
author = {Zijie Gu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26970},
eprint = {2610.26970},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.26970},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Gaussian Lower Bounds for Distinct Subset Sums in Higher Dimensions
AU - Zijie Gu
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2610.26970
UR - https://arxiv.org/abs/2610.26970
AB - For sequences of n nonnegative integer vectors in ℤ^k with distinct subset sums, we prove, for fixed k, the uniform lower bound M≥ (C_k−o(1))2^n/k/√n for the largest coordinate, where C_k=√(2k/π) ((k+1)^(k−1)/(2k))/(√(⌊ (k+1)^2/4⌋)). The proof compares a Gaussian-kernel expectation with a sum over a translated lattice, uniformly over the vector sequence. We optimise the resulting quadratic-form constant using a vertex second-moment certificate. We show that C_k also exceeds the variance constant obtained by optimising an ellipsoidal geometry.
ER -