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Polona Durcik

Publications and source records attributed to Polona Durcik.

29 records · Page 2Linked to original sources
arXiv2020

Boxes, extended boxes, and sets of positive upper density in the Euclidean space

We prove that sets with positive upper Banach density in sufficiently large dimensions contain congruent copies of all sufficiently large dilates of three specific higher-dimensional patterns. These patterns are: $2^n$ vertices of a fixed $n$-dimensional rectangular box, the same vertices extended with $n$ points completing three-term arithmetic progressions, and the same vertices extended with $n$ points completing three-point corners. Our results provide common generalizations of several Euclidean density theorems from the literature.

math.CO↗
arXiv2020

Boundedness of some multi-parameter fiber-wise multiplier operators

We prove $L^p$ estimates for various multi-parameter bi- and trilinear operators with symbols acting on fibers of the two-dimensional functions. In particular, this yields estimates for the general bi-parameter form of the twisted paraproduct studied in arXiv:1011.6140.

math.CA↗
arXiv2019

Directional maximal function along the primes

We study a two-dimensional discrete directional maximal operator along the set of the prime numbers. We show existence of a set of vectors, which are lattice points in a sufficiently large annulus, for which the $\ell^2$ norm of the associated maximal operator with supremum taken over all large scales grows with an epsilon power in the number of vectors. This paper is a follow-up to a prior work on the discrete directional maximal operator along the integers by the first and third author.

math.CA↗
arXiv2019

Singular Brascamp-Lieb: a survey

We present an overview of results on multi-linear singular integrals in the broader context of Brascamp-Lieb inequalities. This elaborates a lecture given at the inspiring conference on Geometric Aspects of Harmonic Analysis at Cortona 2018 in honor of Fulvio Ricci.

math.CA↗
arXiv2019

A polynomial Roth theorem on the real line

For a polynomial $P$ of degree greater than one, we show the existence of patterns of the form $(x,x+t,x+P(t))$ with a gap estimate on $t$ in positive density subsets of the reals. This is an extension of an earlier result of Bourgain. Our proof is a combination of Bourgain's approach and more recent methods that were originally developed for the study of the bilinear Hilbert transform along curves.

math.CO↗
arXiv2017

Norm-variation of ergodic averages with respect to two commuting transformations

We study double ergodic averages with respect to two general commuting transformations and establish a sharp quantitative result on their convergence in the norm. We approach the problem via real harmonic analysis, using recently developed methods for bounding multilinear singular integrals with certain entangled structure. A byproduct of our proof is a bound for a two-dimensional bilinear square function related to the so-called triangular Hilbert transform.

math.DS↗
arXiv2017

On side lengths of corners in positive density subsets of the Euclidean space

We generalize a result by Cook, Magyar, and Pramanik [3] on three-term arithmetic progressions in subsets of $\mathbb{R}^d$ to corners in subsets of $\mathbb{R}^d\times\mathbb{R}^d$. More precisely, if $1<p<\infty$, $p\neq 2$, and $d$ is large enough, we show that an arbitrary measurable set $A\subseteq\mathbb{R}^d\times\mathbb{R}^d$ of positive upper Banach density contains corners $(x,y)$, $(x+s,y)$, $(x,y+s)$ such that the $\ell^p$-norm of the side $s$ attains all sufficiently large real values. Even though we closely follow the basic steps from [3], the proof diverges at the part relying on harmonic analysis. We need to apply a higher-dimensional variant of a multilinear estimate from [5], which we establish using the techniques from [5] and [6].

math.CO↗
arXiv2015

An $L^4$ estimate for a singular entangled quadrilinear form

The twisted paraproduct can be viewed as a two-dimensional trilinear form which appeared in the work by Demeter and Thiele on the two-dimensional bilinear Hilbert transform. $L^p$ boundedness of the twisted paraproduct is due to Kovač, who in parallel established estimates for the dyadic model of a closely related quadrilinear form. We prove an $(L^4,L^4,L^4,L^4)$ bound for the continuous model of the latter by adapting the technique of Kovač to the continuous setting. The mentioned forms belong to a larger class of operators with general modulation invariance. Another instance of such is the triangular Hilbert transform, which controls issues related to two commuting transformations in ergodic theory, and for which $L^p$ bounds remain an open problem.

math.CA↗