Decomposition of random walk measures on the one-dimensional torus

Decomposition of random walk measures on the one-dimensional torus, Discrete Analysis 2020:3, 24 pp.

Let $\mathbb T$ be the one-dimensional torus $\mathbb R/\mathbb Z$. Given a positive integer $n$, we can define a dilation map $T_n$ that multiplies each $x\in\mathbb T$ by $n$ (and reduces mod 1). If $S\subset\mathbb N$, then we say that a measure $\mu$ on $\mathbb T$ is $S$-invariant if $\mu(T_n^{-1}(A))=\mu(A)$ for every $n\in S$ and every $\mu$-measurable set $A$.

A famous conjecture of Furstenberg from 1967 asks for a classification of the invariant measures in the case that $S$ is the multiplicative semigroup of $\mathbb N$ generated by $2$ and $3$. That is, which measures are invariant under all dilations by positive integers that are a product of a power of 2 and a power of 3? Furstenberg conjectured that all such measures are either atomic or equal to Lebesgue measure. (An example of an invariant atomic measure is one that assigns equal weight to all multiples of 1/6.) While there have been many interesting partial results towards this conjecture, it remains open.

This paper is aimed at a more general understanding of how measures on the torus relate to sets of dilations. Given a large integer $L$, the author defines a subset $S\subset[L,2L]$ to be regular if no subinterval of $[L,2L]$ contains too many points of $S$ (the precise condition is given in Definition 1 of the paper), and shows that if $S$ is any set that is both regular and not too small, then an arbitrary measure $\mu$ on $\mathbb T$ can be decomposed into two disjointly supported parts, one of which “spreads out rapidly” when a random dilation from $S$ is applied, and the other of which is concentrated on a union of small and well-separated intervals.

The sense in which a measure spreads out rapidly is the following. Given a measure $\mu$ and a finite subset $S\subset\mathbb N$, we can replace $\mu$ by the measure $\mu_S(A)=|S|^{-1}\sum_{s\in S}\mu(T_s^{-1}(A))$. If we apply this averaging process repeatedly, then after a short time all Fourier coefficients of the resulting measure at small non-zero integers are small, so in a certain useful sense (familiar to number theorists wishing to prove equidistribution results) the measure is close to uniform.

The proof uses tools from additive combinatorics, geometric measure theory, and harmonic analysis.