For every β ∈ (0,∞), β ≠ 1, we prove that a positive measure subset A of the unit square contains a point (x0, y0) such that A nontrivially intersects curves y − y0 = a(x −x0)β for a whole interval I ⊆ (0,∞) of parameters a ∈ I . A classical Nikodym set counterexample prevents one to take 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 Bourgainstyle large-set variants of a recent continuous-parameter Sárközy-type theorem by Kuca, Orponen, and Sahlsten.
Durcik, P., Kovač, V. & Stipčić, M. A Strong-Type Furstenberg–Sárközy Theorem for Sets of Positive Measure. J Geom Anal 33, 255 (2023). https://doi.org/10.1007/s12220-023-01309-7
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