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Creators/Authors contains: "Durcik, Polona"

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  1. ; (, Mathematical Research Letters)
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  2. ; ; (, The Journal of Geometric Analysis)
    Abstract For 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. 
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  3. ; ; (, Mathematische Zeitschrift)
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  4. ; ; ; (, Ergodic Theory and Dynamical Systems)
    Abstract We prove almost everywhere convergence of continuous-time quadratic averages with respect to two commuting $$\mathbb {R}$$ -actions, coming from a single jointly measurable measure-preserving $$\mathbb {R}^2$$ -action on a probability space. The key ingredient of the proof comes from recent work on multilinear singular integrals; more specifically, from the study of a curved model for the triangular Hilbert transform. 
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