Activities This Week

Analysis of Boolean functions aims at “hearing the shape” of functions on the discrete cube {-1,1}^n – namely, at understanding what the structure of the (discrete) Fourier transform tells us about the function. In this talk, we focus on the structure of “low-degree” functions on the discrete cube, namely, on functions whose Fourier coefficients are concentrated on “low” frequencies. While such functions look very simple, we are surprisingly far from understanding them well, even in the most basic first-degree case. We shall present several results on first-degree functions on the discrete cube, including the recent proof of Tomaszewski’s conjecture (1986) which asserts that any first-degree function (viewed as a random variable) lies within one standard deviation from its mean with probability at least 1/2. Then we shall discuss several core open questions, which boil down to understanding, what does the knowledge that a low-degree function is bounded, or is two-valued, tell us about its structure.

Based on joint work with Ohad Klein

Classically, Diophantine approximation deals with the problem of studying “good” approximations of a real number by rational numbers. I will explain the meaning of “good approximants” and the classical main results in this area of research. In particular, Klaus Roth was awarded with the Fields medal in 1955 for proving that the approximation exponent of a real algebraic number is 2. I will present a recent extension of Roth’s theorem in the framework of adelic curves. These mathematical objects, introduced by Chen and Moriwaki in 2020, stand as a generalisation of global fields.

In this talk, we study the density properties in the real line of oscillating sequences of the form $( g(k) \cdot F(kα) )_{k \in \mathbb{N}}$, where $g$ is a positive increasing function and $F$ a real continuous $1$-periodic function. This extends work by Berend, Boshernitzan and Kolesnik who established differential properties on the function F ensuring that the oscillating sequence is dense modulo 1.

More precisely, when $F$ has finitely many roots in $[0,1)$, we provide necessary and sufficient conditions for the oscillating sequence under consideration to be dense in $\mathbb{R}$. All the related results are stated in terms of the Diophantine properties of $α$, with the help of the theory of continued fractions.