Measure and dimension of sums and products

Publication Date

1-1-2021

Document Type

Article

Publication Title

Proceedings of the American Mathematical Society

Volume

149

Issue

9

DOI

10.1090/proc/15513

First Page

3765

Last Page

3780

Abstract

We investigate the Fourier dimension, Hausdorff dimension, and Lebesgue measure of sets of the form RY + Z, where R is a set of scalars and Y, Z are subsets of Euclidean space. Regarding the Fourier dimension, we prove that for each α ∈ [0, 1] and for each non-empty compact set of scalars R ⊆ (0, ∞), there exists a compact set Y ⊆ [1, 2] such that dimF (Y ) = dimH(Y ) = dimM(Y ) = α and dimF (RY ) ≥ min{1, dimF (R) + dimF (Y )}. This theorem verifies a weak form of a more general conjecture, and it can be used to produce new Salem sets from old ones. Further, we investigate lower bounds on the measure and dimension of RY + Z for R ⊂ (0, ∞) and arbitrary Y and Z; these latter results provide a generalized variant of some theorems of Wolff and Oberlin in which Y is the unit sphere.

Keywords

Fourier dimension, Fractals, Hausdorff dimension, Minkowski product, Minkowski sum

Department

Mathematics and Statistics

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