Around Furstenberg’s times $p$, times $q$ conjecture: times $p$-invariant measures with some large Fourier coefficients

Around Furstenberg’s times $p$, times $q$ conjecture: times $p$-invariant measures with some large Fourier coefficients, Discrete Analysis 2024:10, 31 pp.

In the 1960s, Furstenberg proved several results on the following theme. Let $p$ and $q$ be integers that are multiplicatively independent, which means that $\log p/\log q$ is irrational, or equivalently that no power of $p$ is equal to a power of $q$. Then if $x$ is an irrational number, we expect its base-$p$ representation and its base-$q$ representation to have no common structure. An example of a rigorous result in this direction is the following landmark theorem of Furstenberg. Let $\mathbb T$ denote the circle $\mathbb R/\mathbb Z$ and let $T_m:\mathbb T\to\mathbb T$ denote multiplication by $m$ mod 1.

Theorem. If $p$ and $q$ are multiplicatively independent, then the only infinite closed subset of $\mathbb T$ that is invariant under both $T_p$ and $T_q$ is $\mathbb T$.

To see why the hypotheses make sense, note that if $X$ consists of all multiples of $1/m$ for some $m$, then it is closed and invariant under $T_p$ and $T_q$, but finite, and if $X$ consists of all multiples of some irrational number $x$, then it is infinite and invariant under $T_p$ and $T_q$ but not closed. And finally, if $p^m=q^n$ and $d=\mathop{\text{gcd}}(m,n)$, then $p^{m/d}=q^{n/d}$, so we may assume that $m$ and $n$ are coprime. But $p^{m/n}$ is an integer, so $p^m$ must be a perfect $n$th power, from which it follows that $p$ is a perfect $n$th power, and similarly $q$ is a perfect $m$th power. We thus find $r$ such that $p=r^n$ and $q=r^m$. If we then let $a_1<a_2<\dots$ be a rapidly increasing sequence of integers and set $x=\sum_{i=1}^\infty r^{-a_i}$, then the closure of $\{x\}$ under the operations $T_p$ and $T_q$ will be an infinite set with all its elements close to a negative power of $r$, so its closure will not be the whole of $\mathbb T$. Note that such an $x$ has a highly atypical base-$p$ expansion and also a highly atypical base-$q$ expansion, exactly the phenomenon we do not expect to be possible when $p$ and $q$ are multiplicatively independent. And indeed, an immediate consequence of Furstenberg’s theorem is that if $x$ is irrational, then the the numbers of the form $p^mq^nx$ (reduced mod 1) are dense in $\mathbb T$, so at least to that extent our expectation is correct.

Furstenberg went on to make three more precise conjectures. The first was that, under the same assumptions, if $X$ is $T_p$-invariant, then the images $T_q^nX$ converge to $\mathbb T$ in the Hausdorff metric, which means that for every $\epsilon>0$, if $n$ is sufficiently large then every point in $\mathbb T$ is within $\epsilon$ of a point in $T_q^nX$. It is easy to see that this implies the theorem above.

The second was a measure-theoretic version of the theorem, which asserts that if $\mu$ is a probability measure on $\mathbb T$ that is invariant under $T_p$ and $T_q$, then $\mu$ is a convex combination of an atomic measure and Lebesgue measure. (As a moderately interesting example of such a measure, let $r$ is a prime and let $G$ be the multiplicative subgroup of $\mathbb F_r^*$ generated by $p$ and $q$. Then the uniform measure on $G$ is invariant under $T_p$ and $T_q$, as is any convex combination of that measure with Lebesgue measure.)

The third relates to the second in a similar way to how the first relates to the theorem. It states that if $\mu$ is an non-atomic $T_p$-invariant probability measure on $\mathbb T$, then the measures $T_q^n\mu$ converge to Lebesgue measure in the weak-star topology. This means that for every continuous function $f$ defined on $\mathbb T$, the integral of $f$ with respect to $T_q^n\mu$ converges to the integral of $f$ with respect to Lebesgue measure.

The main aim of this paper is to disprove this last conjecture, thus shedding interesting light on what a proof of the $\times p,\times q$ conjecture could be like. In fact, the authors disprove it in a very strong way, by showing not just that there is a counterexample, but by showing that a typical non-atomic $T_p$-invariant probability measure is a counterexample, in the Baire-category sense. That is, they show that the set of non-atomic $T_p$-invariant probability measures $\mu$ for which $T_q^n\mu$ converges to Lebesgue measure in the weak-star topology is meagre (meaning that it is contained in a countable union of nowhere dense sets).

The authors prove this by showing that
$$\limsup_{n\to\infty}|\hat\mu(q^n)|>0$$
for all but a meagre set of non-atomic $T_p$-invariant measures $\mu$. Note that if $\mu$ has this property, then $\int\exp(2\pi i x/q)\mathrm{d}T_q^n\mu$ does not tend to zero, whereas $\int\exp(2\pi ix/q)\mathrm{d}\lambda=0$.

A natural class of examples of $T_p$-invariant measures is given by letting $A$ be a subset of $\{0,1,\dots,p-1\}$ and taking the obvious measure on the set of all numbers whose base-$p$ representation have all their digits in $A$, where the digits are independent and uniformly distributed in $A$. (More generally one could take any non-uniform distribution on $\{0,1,\dots,p-1\}$.) It is not hard to see that such sets will have large Fourier coefficients at $p^n$. Bearing this kind of example in mind, it might seem rather strange that one can use the $T_p$-invariance of $\mu$ to say something about the Fourier coefficients $\hat\mu(q^n)$ for some $q$ that is multiplicatively independent of $p$. The key turns out to be that there is an arithmetic progression of integers $N$ such that the powers of $q$ are eventually periodic mod $p^N-1$. (A clue as to why it is natural to reduce mod $p^N-1$ is that rational numbers with denominator $p^N-1$ are fixed points of $T_p^N$.)

In fact, weaker conditions turn out to suffice: the authors identify a condition they call assumption (H), and show that if $(c_n)$ is any sequence that satisfies (H), then all but a meagre set of non-atomic $T_p$-invariant measures satisfy
$$\limsup_{n\to\infty}|\hat\mu(c_n)|>0.$$
It is not clear whether assumption (H) is necessary: the authors ask the very nice question of whether the same conclusion holds for every integer sequence $(c_n)$ that tends to infinity.

The paper also contains multidimensional generalizations of the main results, as well as other interesting open problems.