Bounds for Sets with No Nontrivial Polynomial Progressions

Abstract

In this thesis, we prove quantitative bounds in the polynomial Szemeredi theorem in several situations in which no bounds were previously known. In Chapter 2, we prove that if P₁, P₂ ∈ Z[y] are affine-linearly independent, then any subset of F_q with no nontrivial polynomial progressions of the form x, x+P₁(y), x+P₂(y) must have size «_{P1, P2}q^23/24, provided the characteristic of Fq is large enough. In Chapter 3, we prove that if P₁,..., P_m 2 Z[y] are affine-linearly independent, then any subset of F_q with no nontrivial polynomial progressions of the form x, x + P₁(y),..., x + P_m(y) must have size «P₁,...,P_m q^{1-γP1,...,Pm} for some γ_P1,...,Pm > 0, again provided that the characteristic of F_q is large enough. In Chapter 4, we prove that any subset of {1,...,N} with no nontrivial progressions of the form x, x+y, x+y² must have size « N=(log logN)^2-157. In Chapter 5, we prove that if P₁,..., P_m 2 Z[y] have distinct degrees, then any subset of {1,...,N} with no nontrivial polynomial progressions of the form x, x + P₁(y),..., x + P_m(y) must have size « N=(log logN)^{1-γ P1,...,Pm} for some P₁,...,P_m > 0. In the nal chapter, Chapter 6, we move to the nonabelian setting and prove power-saving bounds for subsets of nonabelian nite simple groups with no nontrivial progressions of the form x, xy, xy².