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.
Funder
NSF-DMS
Croatian Science Foundation
Grand Challenges Initiative at Chapman University
Publisher
Springer Science and Business Media LLC