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A Generalisation of Maximal (k,b)-Linear-Free Sets of Integers
Abstract
Fix integers k,b,q with k≥ 2,b≥ 0,q≥ 2. Define the function p to be: p(x)=kx+b. We call a set S of integers (k,b,q)-linear-free if x∈ S implies pi (x)∉ S for all i=1,2,…,q−1, where pi (x)=p(pi−1(x)) and p0 (x)=x. Such a set S is maximal in [n]:={1,2,…,n} if S∪ {t},∀ t∈ [n]backslash S is not (k,b,q)-linear-free. Let Mk,b,q (n) be the set of all maximal (k,b,q)-linear-free subsets of [n], and define gk,b,q (n)=minS∈ Mk,b,q (n)|S| and fk,b,q (n)=maxS∈ Mk,b,q (n)|S|. In this paper, formulae for gk,b,q (n) and fk,b,q (n) are proposed. Also, it is proven that there is at least one maximal (k,b,q)-linear-free subset of [n] with exactly x elements for any integer x between gk,b,q (n) and fk,b,q (n), inclusively.
Versions
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BibTeX
@misc{nguyen2026generalisation,
title = {{A Generalisation of Maximal $(k,b)$-Linear-Free Sets of Integers}},
author = {Minh Quang Nguyên},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.17188},
eprint = {2610.17188},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.17188},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Generalisation of Maximal (k,b)-Linear-Free Sets of Integers
AU - Minh Quang Nguyên
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.17188
UR - https://arxiv.org/abs/2610.17188
AB - Fix integers k,b,q with k≥ 2,b≥ 0,q≥ 2. Define the function p to be: p(x)=kx+b. We call a set S of integers (k,b,q)-linear-free if x∈ S implies p^i (x)∉ S for all i=1,2,…,q−1, where p^i (x)=p(p^i−1(x)) and p^0 (x)=x. Such a set S is maximal in [n]:={1,2,…,n} if S∪ {t},∀ t∈ [n]backslash S is not (k,b,q)-linear-free. Let M_k,b,q (n) be the set of all maximal (k,b,q)-linear-free subsets of [n], and define g_k,b,q (n)=min_S∈ M_k,b,q (n)|S| and f_k,b,q (n)=max_S∈ M_k,b,q (n)|S|. In this paper, formulae for g_k,b,q (n) and f_k,b,q (n) are proposed. Also, it is proven that there is at least one maximal (k,b,q)-linear-free subset of [n] with exactly x elements for any integer x between g_k,b,q (n) and f_k,b,q (n), inclusively.
ER -