TL;DR
This paper studies the density of function values on lattice translates, providing new insights into classical problems like Minkowski's conjecture and inhomogeneous Diophantine approximation.
Contribution
It introduces a general framework for analyzing the density of function values on lattice translates under natural assumptions, with applications to longstanding mathematical conjectures.
Findings
Almost all lattice translates yield dense value sets
New partial results on Minkowski's conjecture
Advances in inhomogeneous Diophantine approximation
Abstract
Given a continuous function from Euclidean space to the real line, we analyze (under some natural assumption on the function), the set of values it takes on translates of lattices. Our results are of the flavor: For almost any translate, the set of values is dense in the set of possible values. The results are then applied to a variety of concrete examples, obtaining new information in classical discussions in different areas in mathematics; in particular, Minkowski's conjecture regarding products of inhomogeneous forms and inhomogeneous Diophantine approximations.
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