Fourier dimension and avoidance of linear patterns

DOI10.1016/J.AIM.2022.108252zbMATH Open1497.42011arXiv2006.10941OpenAlexW4213332197MaRDI QIDQ2118908

Malabika Pramanik, Yi Yu Liang

Publication date: 23 March 2022

Published in: Advances in Mathematics (Search for Journal in Brave)

Abstract: The results in this paper are of two types. On one hand, we construct sets of large Fourier dimension that avoid nontrivial solutions of certain classes of linear equations. In particular, given any finite collection of translation-invariant linear equations of the form �egin{equation} sum_{i=1}^v m_ix_i=m_0x_0, ; ext{ with } (m_0, m_1, cdots, m_v) in mathbb N^{v+1}, m_0 = sum_{i=1}^{v} m_i ext{ and } v geq 2, label{rational-eqn} end{equation} we find a Salem set $Esubseteq[0,1]$ of dimension 1 that contains no nontrivial solution of any of these equations; in other words, there does not exist a vector $(x_0,x_1,cdots,x_v)in{E}^{v+1}$ with distinct entries that satisfies any of the given equations. Variants of this construction can also be used to obtain Salem sets that avoid solutions of translation-invariant linear equations of other kinds, for instance, when the collection of linear equations to be avoided is uncountable or has irrational coefficients. While such constructions seem to suggest that Salem sets can avoid many configurations, our second type of results offers a counterpoint. We show that a set in $mathbbR$ whose Fourier dimension exceeds $2/(v+1)$ cannot avoid nontrivial solutions of all equations of the above form. In particular, a set of positive Fourier dimension must contain a nontrivial linear pattern of the above form for some $v$, and hence cannot be rationally independent. This is in stark contrast with known results cite{M17} that ensure the existence of rationally independent sets of full Hausdorff dimension. The latter class of results may be viewed as quantitative evidence of the structural richness of Salem sets of positive dimension, even if the dimension is arbitrarily small.

Full work available at URL: https://arxiv.org/abs/2006.10941

zbMATH Keywords

Fourier dimensionSalem set

Mathematics Subject Classification ID

Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type (42A38) Hausdorff and packing measures (28A78) Trigonometric series of special types (positive coefficients, monotonic coefficients, etc.) (42A32)

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