A strong-type Furstenberg-Sárközy theorem for sets of positive measure

For every $\beta \in (0,\mathrm{\infty })$, $\beta \mathrm{\ne }1$ 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 $I\subseteq (0,\mathrm{\infty })$ of parameters $a\in I$. 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.