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

Abstract

AbstractFor every $$\beta \in (0,\infty )$$ β ∈ ( 0 , ∞ ) , $$\beta \ne 1$$ β ≠ 1 , we prove that a positive measure subset A of the unit square contains a point $$(x_0,y_0)$$ ( x 0 , y 0 ) such that A nontrivially intersects curves $$y-y_0 = a (x-x_0)^\beta $$ y - y 0 = a ( x - x 0 ) β for a whole interval $$I\subseteq (0,\infty )$$ I ⊆ ( 0 , ∞ ) of parameters $$a\in I$$ a ∈ I . A classical Nikodym set counterexample prevents one to take $$\beta =1$$ β = 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.