Bounds for sets with no polynomial progressions

Abstract

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

True complexity of polynomial progressions in finite fields

“…As a consequence, not every term of the progression would be globally controlled by a Gowers norm. We refer the reader to [30][31][32] for an in-depth discussion of these issues.…”

“…, P t (x) 2 are linearly independent, implying that P 2,1 = R t . From (30), it follows that P 1,1 = Ψ [1] and P i,1 ⊆ Ψ [i] for i > 1. Together with the fact that P 2,1 = R t , this implies that Ψ [2] = P 2,1 , and so the groups G P = G Ψ are in fact the same for any group G.…”

A Polylogarithmic Bound in the Nonlinear Roth Theorem

“…, f m ) := E x∈[N ] E y∈[N 1/ deg Pm ] f 0 (x)f 1 (x + P 1 (y)) • • • f m (x + P m (y)). Using the main technical result of [Pel19], [Pel19, Theorem 3.3], one can show that if Λ N P 1 ,...,Pm (f 0 , f 1 , . .…”

“…, Q m , which arise on passing to a subprogression, may not satisfy the hypotheses required to reapply [Pel19, Theorem 3.3]. It is likely that the polynomials are sufficiently well-behaved for the arguments of [Pel19] to remain valid, but we leave this verification to the energetic reader.…”

“…Longer polynomial progressions. In analogy with the first author's generalisation [Pel19] of [PP19], it is natural to ask whether the methods of this paper yield polylogarithmic bounds for sets of integers lacking longer progressions…”