# International Workshop "Dispersion and fixed volume discrepancy"

The laboratory "High-dimensional approximation and applications" in cooperation with Moscow Center of Fundamental and Applied Mathematics and Johannes Kepler University Linz will held an online (via Zoom) one day workshop "Dispersion and fixed volume discrepancy" on December 13. Everyone is invited to attend.

**Registration form (to get Zoom link)**

Program (Moscow time, UTC+3):

Monday, December 13

→ 17:20 Vladimir Temlyakov, "Smooth fixed volume discrepancy, dispersion, and related problems", video;

## Abstract

It will be an introductory type lecture. I'll begin with the classical concepts of numerical integration and discrepancy. Then, I'll proceed to the concepts of smooth discrepancy and smooth fixed volume discrepancy. Results on the smooth fixed volume discrepancy for specific sets of points (Fibonacci, Frolov) will be discussed. Also, application of these results for dispersion will be discussed.→ 18:20 Nastya Rubtsova, "On the fixed volume discrepancy of the Korobov point sets", video;

## Abstract

We will discuss the fixed volume discrepancy of the Korobov point sets in the unit cube. It was observed recently that this new characteristic allows us to obtain optimal rate of dispersion from numerical integration results. So we study this new version of discrepancy, which seems to be interesting by itself. Our work extends recent results by V. Temlyakov and M. Ullrich on the fixed volume discrepancy of the Fibonacci point sets.→ 19:00 Alexander Litvak, "On the minimal dispersion in the unit cube", video;

## Abstract

We improve known upper bounds for the minimal dispersion of a point set in the unit cube and its inverse. Some of our bounds are sharp up to logarithmic factors. The talk is partially based on a joint work with G. Livshyts.→ 20:00 Boris Bukh, "Dispersion in a fixed dimension. Part 1", video;

## Abstract

The dispersion \(\text{disp}(S)\) of an \(n\)-set \(S\) in \([0,1]^d\) is the volume of the largest empty axis-parallel box in \([0,1]^d\). The minimal dispersion \(\text{disp}(n, d)\) is the infimum of \(\text{disp}(S)\) among all possible \(S\). It is known that, for fixed dimension \(d\), \(\text{disp}(n, d) \sim c_d/n\) when \(n\) is large. We prove that \(\Omega(d)\leq c_d\leq O(d^2 \log{d})\). Furtheremore, the set attaining the upper bound can be generated in linear time. We use the same ideas to construct digital almost nets, which are an approximate version of \((t,m,s)\)-nets, that hit all the dyadic boxes of proper volume not just once but approximately the expected number many times.In part I, we shall describe most of the results, and explain the lower bounds. Part II will be primarily devoted to constructions.

→ 21:00 Ting-Wei Chao, "Dispersion in a fixed dimension. Part 2", video.

## Abstract

The dispersion \(\text{disp}(S)\) of an \(n\)-set \(S\) in \([0,1]^d\) is the volume of the largest empty axis-parallel box in \([0,1]^d\). The minimal dispersion \(\text{disp}(n, d)\) is the infimum of \(\text{disp}(S)\) among all possible \(S\). It is known that, for fixed dimension \(d\), \(\text{disp}(n, d) \sim c_d/n\) when \(n\) is large. We prove that \(\Omega(d)\leq c_d\leq O(d^2 \log{d})\). Furtheremore, the set attaining the upper bound can be generated in linear time. We use the same ideas to construct digital almost nets, which are an approximate version of \((t,m,s)\)-nets, that hit all the dyadic boxes of proper volume not just once but approximately the expected number many times.In part I, we shall describe most of the results, and explain the lower bounds. Part II will be primarily devoted to constructions.

Organizing commitee:

→ V.N. Temlyakov (chair)

→ Mario Ullrich (chair)

→ E.D. Kosov (scientific secretary)

→ V.E. Podolskii

Contacts: approx.lab.msu@gmail.com