Dimension bounds for singular affine forms
Gaurav Aggarwal
TL;DR
The paper develops upper bounds for the Hausdorff and packing dimensions of singular-on-average and $\omega$-singular affine forms in singly metric settings, addressing questions of Das–Fishman–Simmons–Urbański and Kleinbock–Wadleigh. It combines homogeneous dynamics via Dani’s correspondence with a fixed-matrix analysis (using EMass and covering arguments) and a fixed-shift analysis (via height functions and contraction hypotheses) to obtain dimension bounds that extend to generalized weighted setups and fractal intersections. Core innovations include a unified treatment through Div and Div$_{\theta}$, explicit dimension bounds in terms of escape-of-mass quantities $\text{EMass}(\cdot)$, and a height-function framework that handles near-homogeneous-subspace recurrences in the fractal setting. Applications cover equal-weight cases and intersections with self-similar fractals, yielding sharp bounds and zero-measure/zero-dimension conclusions in many natural fractal measures. The results advance the understanding of Diophantine properties of affine forms under partial metric information and provide tools for analyzing fractal Diophantine phenomena via dynamical methods.
Abstract
In this paper, we establish upper bounds on the dimension of sets of singular-on-average and \(ω\)-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 Urbański, 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.