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The structure of critical sets for F_p arithmetic progressions
Ernie Croot
Fix a prime p and a density 0 < d <= 1. Among all functions f : F_p -> [0,1], what can one say about those which assign minimal weight to three-term arithmetic progressions -- that…
Arithmetic structures in smooth subsets of F_p
Ernie Croot
Fix integers a_1,...,a_d satisfying a_1 + ... + a_d = 0. Suppose that f : Z_N -> [0,1], where N is prime. We show that if f is ``smooth enough'' then we can bound from below the su…
On Sumsets and Spectral Gaps
Ernie Croot, Tomasz Schoen
It is well known that if S is a subset of the integers mod p, and if the second-largest Fourier coefficient is ``small'' relative to the largest coefficient, then the sumset S+S is…
An application of linear programming duality to discrete Fourier analysis and additive problems
Ernie Croot
Suppose that f is a function from Z_p -> [0,1] (Z_p is my notation for the integers mod p, not the p-adics), and suppose that a_1,...,a_k are some places in Z_p. In some additive n…
Subsets of F_p^n without three term arithmetic progressions have several large Fourier coefficients
Ernie Croot
Suppose that f : F_p^n -> [0,1] has expected value t in [p^(-n/9),1] (so, the density t can be quite low!). Furthermore, suppose that support(f) has no three-term arithmetic progre…