Abstract
It is shown that if and if is a set of upper density , then—in a sense depending on —all large dilates of any given k-dimensional simplex can be embedded in . A simplex can be embedded in the set if contains simplex , which is isometric to . Moreover, the same is true if only is assumed, and satisfies some immediate necessary conditions.
The proof uses techniques of harmonic analysis developed for the continuous case, as well as a variant of the circle method due to Siegel [S]
Citation
Ákos Magyar. "-Point configurations in sets of positive density of ." Duke Math. J. 146 (1) 1 - 34, 15 January 2009. https://doi.org/10.1215/00127094-2008-060
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