A curved three-point pattern problem for fractal sets on the real line

We study the occurrence of curved three-point configurations in fractal subsets of the real line. We prove that if $E\subset [0,1]$ is a compact set with sufficiently large Hausdorff dimension, then $E$ contains a curved three-point progression associated with a broad class of nonlinear functions. Our approach can also show the existence of the curved three-point pattern under the assumption that the Hausdorff content of $E$ is bounded away from zero. The class of functions includes, in addition to polynomials with vanishing constant term, nonlinear functions such as $$t^k\log (1+t),\mathrm{\forall }k\ge 1.$$