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Corners with polynomial side length
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Corners with polynomial side length
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A $P$-polynomial corner, for $P \in \mathbb{Z}[z]$ a polynomial, is a triple of points $(x,y),\; (x+P(z),y),\; (x,y+P(z))$ for $x,y,z \in \mathbb{Z}$. In the case where $P$ has an integer root of multiplicity $1$, we show that if $A \subseteq [N]^2$ does not contain any nontrivial $P$-polynomial corners, then $$|A| \ll_P \frac{N^2}{(\log\log\log N)^c}$$ for some absolute constant $c>0$. 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 $z^2-1$. The main ingredients in our proof are a multidimensional quantitative concatenation result from our companion paper arXiv:2407.08636 and a novel degree-lowering argument for box norms.
Forward citations
Cited by 3 Pith papers
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On hyperbolic corners and unit-area triangles in planar sets of large measure
Measurable sets in [0,R]² avoiding upward right triangles of area 1/2 satisfy |A| = O_c(R²/(log R)^c) for c<1/4 with Ω(R log R) example; for fixed-area triangles the bound sharpens to c<1/2 using a hyperbolic trilinea...
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A multidimensional Szemer\'{e}di theorem in integers
Dense subsets of [N]^n contain configurations x, x + r^{m1}e1, ..., x + r^{mn}en for any fixed n and increasing exponents m_i, with density threshold (log N)^{-c}.
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Polynomial Corners Over finite Fields
In (F_p)^2, any set avoiding the polynomial corner configuration has density o(1) as p grows, with a bound stronger than the corresponding integer result under stated conditions on P.
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