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.