Brascamp--Lieb inequalities for fractal dimensions

TL;DR

This paper applies the Brascamp--Lieb inequality to derive new bounds for fractal dimensions like upper box, packing, and Assouad dimensions, revealing limitations for Hausdorff and lower box dimensions.

Contribution

It introduces novel inequalities relating fractal dimensions to projections and sumsets, expanding the application of Brascamp--Lieb inequalities in fractal geometry.

Findings

Derived inequalities for upper box, packing, and Assouad dimensions.

Established new exceptional set estimates for orthogonal projections.

Provided sharp dimension bounds for constrained sumsets.

Abstract

We use the Brascamp--Lieb inequality from functional analysis to prove novel inequalities for the upper box, packing, and Assouad dimensions of fractal sets in terms of the dimensions of certain projections. Analogous inequalities do not hold for Hausdorff or lower box dimensions. We apply these fractal Brascamp--Lieb inequalities to establish new exceptional set estimates for orthogonal projections and to provide sharp dimension estimates for certain constrained sumsets. We also establish analogous nonlinear inequalities via the nonlinear Brascamp--Lieb inequality.