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A Non-Linear Roth Theorem for Fractals of Sufficiently Large Dimension
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abstract
Suppose that $d \geq 2$, and that $A \subset [0,1]$ has sufficiently large dimension, $1 - \epsilon_d < \dim_H(A) < 1$. Then for any polynomial $P$ of degree $d$ with no constant term, there exists a point configuration $\{ x, x-t,x-P(t) \} \subset A$ with $t \approx_P 1$.
Forward citations
Cited by 2 Pith papers
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Polynomial Szemer\'edi for sets with large Hausdorff dimension on the Torus
On the torus, Hausdorff dimension above a threshold 1-epsilon forces arbitrary-length polynomial progressions for distinct-degree polynomials with zero constant terms.
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On Multi-linear Maximal Operators Along Homogeneous Curves
Establishes L^p bound for multi-linear maximal operator along homogeneous polynomial curves under exponent conditions.
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