arXiv subjects

Mario Stipčić

Publications and source records attributed to Mario Stipčić.

4 recordsLinked to original sources

arXiv2023

A strong-type Furstenberg-Sárközy theorem for sets of positive measure

For every $β\in(0,\infty)$, $β\neq 1$ 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 $y-y_0 = a (x-x_0)^β$ for a whole interval $I\subseteq(0,\infty)$ of parameters $a\in I$. 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 Bourgain-style large-set variants of a recent continuous-parameter Sárközy-type theorem by Kuca, Orponen, and Sahlsten.

math.CA↗

arXiv2022

Quantitative bounds for product of simplices in subsets of the unit cube

For each $1\leq i \le n$, let $k_i\geq 1$ and let $Δ_i$ be a set of vertices of a non-degenerate simplex of $k_i+1$ points in $\mathbb{R}^{k_i+1}$. If $A\subseteq [0,1]^{k_1+1}\times \cdots \times [0,1]^{k_n+1}$ is a Lebesgue measurable set of measure at least $δ$, we show that there exists an interval $I=I(Δ_1,\ldots, Δ_n,A)$ of length at least $\exp(-δ^{-C(Δ_1,\ldots, Δ_n)})$ such that for each $λ\in I$, the set $A$ contains $Δ'_1\times \cdots \times Δ'_n$, where each $Δ_i'$ is an isometric copy of $λΔ_i$. This is a quantitative improvement of a result by Lyall and Magyar. Our proof relies on harmonic analysis. The main ingredient in the proof are cancellation estimates for forms similar to multilinear singular integrals associated with $n$-partite $n$-regular hypergraphs.

math.CO↗

arXiv2020

Convergence of ergodic-martingale paraproducts

In this note we introduce a sequence of bilinear operators that unify ergodic averages and backward martingales in a nontrivial way. We establish its convergence in a range of $L^p$-norms and leave its a.s. convergence as an open problem. This problem shares some similarities with a well-known unresolved conjecture on a.s. convergence of double ergodic averages with respect to two commuting transformations.

math.PR↗

arXiv2019

T(1) theorem for dyadic singular integral forms associated with hypergraphs

This paper studies dyadic singular integral forms associated with $r$-partite $r$-uniform hypergraphs such that all their connected components are complete. We characterize their $L^p$ boundedness by T(1)-type conditions in two different ways. We also dominate these forms by positive sparse forms and prove weighted estimates with multilinear Muckenhoupt weights.

math.CA↗