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).
Mattila,Geometry of sets and measures in Euclidean spaces, Cambridge Studies in Advanced Math- ematics, 44, Cambridge Univ
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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).