Dimension bounds for singular affine forms

TL;DR

This paper establishes upper bounds on the dimension of certain singular affine form sets in metric spaces, addressing open questions and extending results to weighted and fractal contexts.

Contribution

It provides new upper bounds for singular affine form sets, partially answering open problems and extending to weighted and fractal scenarios.

Findings

Derived upper bounds for singular affine form sets in metric spaces.

Extended results to weighted settings and fractal intersections.

Addressed open questions in the theory of singular affine forms.

Abstract

In this paper, we establish upper bounds on the dimension of sets of singular-on-average and \(\omega\)-singular affine forms in singly metric settings, where either the matrix or the shift is fixed. These results partially address open questions posed by Das, Fishman, Simmons, and Urba\'nski, as well as Kleinbock and Wadleigh. Furthermore, we extend our results to the generalized weighted setup and derive bounds for the intersection of these sets with a wide class of fractals.