Abstract
We discuss a new direction in which the use of some methods from arithmetic combinatorics can be extended. We consider functions taking values in Euclidean space and supported on subsets of {1, 2, . . ., N}. In this context we present a proof of a natural generalization of Szemerédi's theorem. We also prove a similar generalization of a theorem of Sárkőzy using a vector-valued Fourier transform, adapting an argument of Green and obtaining effective bounds.
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