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Commentarii Mathematici Helvetici

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Volume 88, Issue 2, 2013, pp. 485–506
DOI: 10.4171/CMH/293

Published online: 2013-04-24

Grids with dense values

Uri Shapira[1]

(1) Technion - Israel Institute of Technology, Haifa, Israel

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.

Keywords: Lattices, grids, Minkowski’s conjecture, non-singular forms, Diophantine approximation

Shapira Uri: Grids with dense values. Comment. Math. Helv. 88 (2013), 485-506. doi: 10.4171/CMH/293