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Bounding the Local Dimension of the Convolution of Measures

Kevin G. Hare, Joaquin G. Prandi

TL;DR

This work investigates the local dimension of the convolution of two finite measures on groups and the real line. The central contribution is a general upper bound: if the upper local dimension of $oldsymbol{ u}$ satisfies $ar{ ext{dim}}_{loc}oldsymbol{ u}(x) leq \lambda$ on the interior of its support, then for any regular measure $oldsymbol{ ho}$, $ar{ ext{dim}}_{loc}(oldsymbol{ u}*oldsymbol{ ho})(z) leq \lambda$ for all $z$ in the interior of the product of supports; this is established via a group-theoretic argument using sets $M_z$, $N_z$ and translation maps. The paper extends the result to the real line with gap conditions on the supports, analyzes local dimensions at special boundary points, and discusses sharpness and limitations through concrete examples. These results advance understanding of how convolution re-distributes mass and how interior points versus boundary points control local dimensions, with implications for multifractal analysis and potential generalizations to higher dimensions. The findings also raise open questions about weaker hypotheses for lower bounds and extensions to $ ext{R}^n$ where gap geometry becomes more complex.

Abstract

We study the local dimension of the convolution of two measures. We give conditions for bounding the local dimension of the convolution on the basis of the local dimension of one of them. Moreover, we give a formula for the local dimension of some special points in the support of the convolution.

Bounding the Local Dimension of the Convolution of Measures

TL;DR

This work investigates the local dimension of the convolution of two finite measures on groups and the real line. The central contribution is a general upper bound: if the upper local dimension of satisfies on the interior of its support, then for any regular measure , for all in the interior of the product of supports; this is established via a group-theoretic argument using sets , and translation maps. The paper extends the result to the real line with gap conditions on the supports, analyzes local dimensions at special boundary points, and discusses sharpness and limitations through concrete examples. These results advance understanding of how convolution re-distributes mass and how interior points versus boundary points control local dimensions, with implications for multifractal analysis and potential generalizations to higher dimensions. The findings also raise open questions about weaker hypotheses for lower bounds and extensions to where gap geometry becomes more complex.

Abstract

We study the local dimension of the convolution of two measures. We give conditions for bounding the local dimension of the convolution on the basis of the local dimension of one of them. Moreover, we give a formula for the local dimension of some special points in the support of the convolution.

Paper Structure

This paper contains 6 sections, 21 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. Basic results and definitions
  3. Topological Groups
  4. The Real Line
  5. Local Dimension at Special Points
  6. Conclusions and Open Questions

Key Result

Theorem 1.1

Let $G$ be a group with a compatible metric $d$ that is translation-invariant. Let $\mu$ and $\nu$ be regular measures. Then if $\overline{\operatorname{dim}}_{loc}\mu(x)\leq\lambda$ for all $x\in \operatorname{int}(\operatorname{supp}\mu)$ then $\overline{\operatorname{dim}}_{loc}(\mu*\nu)(z)\leq \

Theorems & Definitions (46)

  • Theorem 1.1
  • Definition 1.2
  • Definition 1.3
  • Lemma 1.4
  • Theorem
  • proof
  • Remark
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • ...and 36 more