Corners with polynomial side length

We prove “reasonable” quantitative bounds for sets in Z^2 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 z^2-1. 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

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