Metric Diophantine approximation with congruence conditions

We prove a version of the Khinchin–Groshev theorem for Diophantine approximation of matrices subject to a congruence condition. The proof relies on an extension of the Dani correspondence to the quotient by a congruence subgroup. This correspondence together with a multiple ergodic theorem are used to study rational approximations in several congruence classes simultaneously. The result in this part holds in the generality of weighted approximation but is restricted to simple approximation functions.

Explicit Salem sets, Fourier restriction, and metric Diophantine approximation in the p-adic numbers

We exhibit the first explicit examples of Salem sets in ℚp of every dimension 0 < α < 1 by showing that certain sets of well-approximable p-adic numbers are Salem sets. We construct measures supported on these sets that satisfy essentially optimal Fourier decay and upper regularity conditions, and we observe that these conditions imply that the measures satisfy strong Fourier restriction inequalities. We also partially generalize our results to higher dimensions. Our results extend theorems of Kaufman, Papadimitropoulos, and Hambrook from the real to the p-adic setting.

Remarks on Diophantine approximation in function fields

We study some problems in metric Diophantine approximation over local fields of positive characteristic.

Federer Measures, Good and Nonplanar Functions of Metric Diophantine Approximation

The goal of this paper is to generalize the main results of [1] and subsequent papers on metric Diophantine approximation with dependent quantities to the set-up of systems of linear forms. In particular, we establish “joint strong extremality” of arbitrary finite collection of smooth nondegenerate submani- folds of .The proof was based on quantitative nondivergence estimates for quasi-polynomial flows on the space of lattices.