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Bounded asymptotic bases for linear forms
Abstract
For a vector of positive integers b = (b1,…,bh) with gcd(b1,…,bh) = 1, we study sets A ⊆ ℕ for which every sufficiently large integer has a bounded positive number of representations n = b1 x1 + ⋯ + bh xh (x1,…,xh∈ A). We prove that such a set exists for every binary vector b ≠ (1,1), and for some general higher-dimensional families, including b = (u1, pd u2, …, p(h−1)d uh) where pnmid u1⋯ uh.
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BibTeX
@misc{tafula2026bounded,
title = {{Bounded asymptotic bases for linear forms}},
author = {Christian Táfula},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11848},
eprint = {2609.11848},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11848},
}
RIS
TY - UNPB
TI - Bounded asymptotic bases for linear forms
AU - Christian Táfula
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.11848
UR - https://arxiv.org/abs/2609.11848
AB - For a vector of positive integers b = (b_1,…,b_h) with gcd(b_1,…,b_h) = 1, we study sets A ⊆ ℕ for which every sufficiently large integer has a bounded positive number of representations n = b_1 x_1 + ⋯ + b_h x_h (x_1,…,x_h∈ A). We prove that such a set exists for every binary vector b ≠ (1,1), and for some general higher-dimensional families, including b = (u_1, p^d u_2, …, p^(h−1)d u_h) where pnmid u_1⋯ u_h.
ER -