Compact sets on the real line with large Hausdorff dimension contain curved three-point progressions for nonlinear functions such as polynomials with zero constant term and t^k log(1+t).
Bourgain,A nonlinear version of Roth’s theorem for sets of positive density in the real line, J
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A curved three-point pattern problem for fractal sets on the real line
Compact sets on the real line with large Hausdorff dimension contain curved three-point progressions for nonlinear functions such as polynomials with zero constant term and t^k log(1+t).