A Strong-Type Furstenberg–Sárközy Theorem for Sets of Positive Measure

Abstract

For every beta in (0,infty ), beta ne 1, we prove that a positive measure subset A of the unit square contains a point (x_0,y_0) such that A nontrivially intersects curves y-y_0 = a (x-x_0)^beta for a whole interval Isubseteq (0,infty ) of parameters ain I. A classical Nikodym set counterexample prevents one to take beta =1, which is the case of straight lines. Moreover, for a planar set A of positive density, we show that the interval I can be arbitrarily large on the logarithmic scale. These results can be thought of as Bourgain-style large-set variants of a recent continuous-parameter Sárközy-type theorem by Kuca, Orponen, and Sahlsten.