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Bounds for sets with no polynomial progressions

Abstract:

Let P_1,..., P_m ∈ Z[y] be polynomials with distinct degrees, each having zero constant term. We show that any subset A of {1,..., N} with no nontrivial progressions of the form x, x + P_1(y),..., x + P_m(y) has size |A| ≪ N/(log log N)^(cP1,...,Pm) . Along the way, we prove a general result controlling weighted counts of polynomial progressions by Gowers norms.

[-1] https://arxiv.org/pdf/1909.00309.pdf






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