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DIMENSIONS OF SETS AVOIDING APPROXIMATE NONTRIVIAL ZEROS OF LINEAR PATTERNS

Part of: Classical measure theory Sequences and sets

Published online by Cambridge University Press:  21 July 2025

ZHIGANG TIAN Open the ORCID record for ZHIGANG TIAN [Opens in a new window]  and
LEI SHANG Open the ORCID record for LEI SHANG [Opens in a new window]
ZHIGANG TIAN
Affiliation:
School of Mathematics, https://ror.org/0530pts50 South China University of Technology , Guangzhou 510640, PR China e-mail: 202010106099@mail.scut.edu.cn
LEI SHANG*
Affiliation:
College of Sciences, https://ror.org/05td3s095 Nanjing Agricultural University , Nanjing 210095, PR China
*
e-mail: shanglei@njau.edu.cn
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Abstract

Given any finite collection $\mathcal {G}$ of translation-invariant linear equations of the form

$$ \begin{align*} \sum_{i=1}^{v} m_i x_i = m_0 x_0, \end{align*} $$

where $(m_0, m_1, \ldots , m_v) \in \mathbb {N}^{v+1}$, $m_0 = \sum _{i=1}^{v} m_i$ and $v \ge 2$, we estimate lower and upper bounds of the supremum of the Hausdorff dimension of sets on the real line that uniformly avoid nontrivial zeros of any f in $\mathcal {G}$.

Keywords

translation-invariant linear equationsnontrivial zeros of linear patternsHausdorff dimension

MSC classification

Primary: 28A80: Fractals
Secondary: 11B30: Arithmetic combinatorics; higher degree uniformity

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , First View , pp. 1 - 12
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

This work was supported by the Natural Science Foundation of Jiangsu Province (Grant No. BK20241541).

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