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.
arXiv comments
55 pages; v2: included a new theorem controlling general polynomial progressions by Gowers norms; v3: added to the introduction and stated some intermediate results in greater generality; v4: referee suggestions incorporated
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|…
The abstract of the paper (https://arxiv.org/abs/1909.00309) says > Let P1, … , Pm ∈ Z[y] be polynomials with distinct degrees ... progressions of the form x, x+P1(y), … , x+Pm(y) Such progressions are a generalization of the "shifted geometric" progressions described in the…
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|…
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The abstract of the paper (https://arxiv.org/abs/1909.00309) says > Let P1, … , Pm ∈ Z[y] be polynomials with distinct degrees ... progressions of the form x, x+P1(y), … , x+Pm(y) Such progressions are a generalization of the "shifted geometric" progressions described in the…
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