A strong-type Furstenberg-Sárközy theorem for sets of positive measure. (English) Zbl 1519.28003
Summary: 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 Sárközy-type theorem by Kuca, Orponen, and Sahlsten.
MSC:
| 28A75 | Length, area, volume, other geometric measure theory |
| 42B25 | Maximal functions, Littlewood-Paley theory |
References:
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