Corners with Polynomial Side Length

We prove ''reasonable'' quantitative bounds for sets in ℤ2

avoiding the polynomial corner configuration (x,y),(x+P(z),y),(x,y+P(z)), where P is any fixed integer-coefficient polynomial with an integer root of multiplicity 1. This simultaneously generalizes a result of Shkredov about corner-free sets and a recent result of Peluse, Sah, and Sawhney about sets without 3-term arithmetic progressions of common difference z2−1. Two ingredients in our proof are a general quantitative concatenation result for multidimensional polynomial progressions and a new degree-lowering argument for box norms.  Joint work with Borys Kuca and James Leng.