  For every $\beta\in(0,\infty)$, $\beta\neq 1$ 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 $I\subseteq(0,\infty)$ 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 S\'{a}rk\"{o}zy-type theorem by Kuca, Orponen, and
Sahlsten.
