On distance measures for well-distributed sets

Details On distance measures for well-distributed sets On distance measures for well-distributed sets

2006-01-20

arxiv.org/abs/math/0601501v1

Mathematics

Classical Analysis and ODEs

Scientific paper

In this paper we investigate the Erd\"os/Falconer distance conjecture for a natural class of sets statistically, though not necessarily arithmetically, similar to a lattice. We prove a good upper bound for spherical means that have been classically used to study this problem. We conjecture that a majorant for the spherical means suffices to prove the distance conjecture(s) in this setting. For a class of non-Euclidean distances, we show that this generally cannot be achieved, at least in dimension two, by considering integer point distributions on convex curves and surfaces. In higher dimensions, we link this problem to the question about the existence of smooth well-curved hypersurfaces that support many integer points.

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