Sequences whose dilations are uniformly distributed modulo one for every irrational number: the exceptional set of Weyl's metric equidistribution theorem

Weyl's 1916 memoir on uniform distribution contains a metric theorem (Satz 21): for every strictly increasing sequence of integers $(n_k)$, the dilated sequence $(n_k x)$ is uniformly distributed modulo $1$ for almost every real $x$, and the null exceptional set provably contains all rationals. Weyl recorded explicitly that it remained undecided whether this exceptional set contains any irrational numbers; in the same section he strengthened a denseness question of Hardy and Littlewood (1914) to uniform distribution and stated that an exhaustive answer cannot be given. Since rational $x$ always produces dilations taking only finitely many values modulo $1$, and hence never uniformly distributed ones, the exceptional set is as small as possible — exactly the rationals — precisely when $(n_k x)$ is uniformly distributed modulo $1$ for every irrational $x$. The problem is to characterize the strictly increasing integer sequences with this universal property. The boundary is currently known only through extremes: polynomial sequences, the primes and Weingartner's multiplicative families are universal, while lacunary sequences have full-dimensional exceptional sets, Dubickas and Bugeaud constructed slowly increasing sequences with explicit quadratic irrational exceptions, and Berend–Boshernitzan–Kolesnik–Kumar (2024) realized every Hausdorff dimension in $[0,1]$ as a density-exceptional set; the enumerations of Furstenberg's multiplicatively independent semigroups are known to satisfy only the weaker density version of universality; no structural characterization is known.

Background

A sequence of real numbers $(y_n)$ is uniformly distributed modulo $1$ (u.d. mod $1$) if, for every interval $[a,b)\subseteq[0,1)$, the proportion of indices $n\le N$ with fractional part $\{y_n\}$ in $[a,b)$ tends to $b-a$ as $N$ grows. By Weyl's criterion this is equivalent to the vanishing of the exponential sums $\frac{1}{N}\sum_{n\le N}e^{2\pi i h y_n}$ for every nonzero integer $h$. The quality of distribution is quantified by the discrepancy $D_N$, the supremum over intervals of the deviation of the empirical frequency from $b-a$; $D_N\to 0$ forces uniform distribution. For a strictly increasing sequence of integers $(n_k)$ and a real number $x$, the dilation $(n_k x)$ records the orbit of $x$ under multiplication by the $n_k$ on the circle $\mathbb{R}/\mathbb{Z}$.

Weyl's 1916 memoir proved, among other things: the polynomial theorem — $(p(n)x)$ is u.d. mod $1$ whenever $p$ is a polynomial with at least one non-constant irrational coefficient, so in particular $(n^d x)$ is u.d. for every irrational $x$ and every positive integer $d$; and the metric theorem (Satz 21) — if a sequence of real numbers grows not too slowly as a function of its index (a condition automatic for strictly increasing integers), then the dilated sequence is u.d. mod $1$ for every $x$ outside a set of Lebesgue measure zero (the standard reference for this circle of ideas is Kuipers–Niederreiter 1974). For integer sequences the exceptional set $E(n_k)$, defined as the set of real $x$ for which $(n_k x)$ fails to be u.d. mod $1$, always contains the rationals, since for rational $x$ the fractional parts take only finitely many values and hence are not uniformly distributed, and Weyl noted on p. 345 that it 'of course contains all rational numbers, but it remains undecided whether further numbers are contained in it'. At the opening of the memoir's seventh section he attributes to Hardy and Littlewood the denseness question of when an increasing integer sequence $l_1<l_2<\cdots$ has $(l_k\xi)$ coming arbitrarily close to every value mod $1$ for every irrational $\xi$; he immediately strengthens it to uniform distribution, asking when $(l_k\xi)$ is u.d. mod $1$ for every irrational $\xi$, and states that an exhaustive answer cannot be given. For strictly increasing integer sequences, $E(n_k)$ equals the set of rationals exactly when $(n_k x)$ is u.d. mod $1$ for every irrational $x$, since rational points can never be removed from the exceptional set; the characterization problem posed below is Weyl's uniform-distribution strengthening together with the exceptional-set question he left undecided. The density version of this universal property is called being a universally densifying sequence in the recent literature (Katz 2018).

The boundary of the problem is mapped only at its extremes. Universal sequences, those with $E(n_k)$ equal to the rationals, include: polynomial sequences (Weyl); the primes (Vinogradov, via exponential-sum estimates for phases over primes, as documented in Weingartner 2023); and the multiplicative families of Weingartner 2023 (almost primes — integers with a bounded number of prime factors — together with the squarefree, practical, densely divisible and lexicographical numbers; with discrepancy estimates when the irrationality measure of $\alpha$, the supremum of exponents $\mu$ such that $|\alpha-p/q|<q^{-\mu}$ for infinitely many rationals $p/q$, is finite). For the weaker density property — mere denseness of the dilated sequence modulo $1$ — Furstenberg (1967) proved that the increasing enumeration of the multiplicative semigroup $\{q_1^n q_2^m\}$ generated by two multiplicatively independent integers $q_1,q_2$, that is with $\log q_2/\log q_1$ irrational, is dense modulo $1$ for every irrational $x$ (an elementary proof is due to Boshernitzan 1994, and Katz 2018 gave generalizations); equivalently, the density-exceptional set of such semigroup sequences is exactly $\mathbb{Q}$ — a statement about denseness that does not by itself settle uniform distribution for these sequences. Boshernitzan (1994) proved that sublacunary sequences — those with $n_{k+1}/n_k\to 1$ — have density-exceptional set of Hausdorff dimension $0$ (Hausdorff dimension being the standard fractal dimension of a subset of the line, ranging from $0$ for countable sets to $1$ for intervals); note this does not assert that the exceptional set equals the rationals.

Non-universal sequences: lacunary sequences, those satisfying the Hadamard gap condition $n_{k+1}/n_k\ge q>1$, fail maximally — the exceptional set for uniform distribution has Hausdorff dimension $1$ (Erdős–Taylor 1957), as does the exceptional set for the weaker density property (Pollington 1979; de Mathan 1980), and even the intersection over all integer bases $q\ge 2$ of the exceptional sets of the geometric sequences $(q^n)$ has Hausdorff dimension $1$ (Schmidt 1966). For the special family $n_k=q^k$, the exceptional points are exactly the real numbers not normal to base $q$, those whose base-$q$ digit frequencies fail to be asymptotically uniform, so deciding membership of any single explicitly given irrational in such an exceptional set contains the classical problem of normality of explicit constants. Ruzsa (1983) constructed, for every $\beta\ge 1$, sequences $r_k$ bounded by a constant times $k^\beta$ whose u.d.-exceptional set has Hausdorff dimension exactly $1-1/\beta$, matching the general upper bound for sequences of polynomial growth (Erdős–Taylor 1957; see Berend–Boshernitzan–Kolesnik–Kumar 2024 for the chain of attribution). Dubickas (2009) and Bugeaud (2009) produced slowly increasing integer sequences whose exceptional sets contain irrationals — in Dubickas's construction, a quadratic irrational $\alpha$ (an irrational root of a quadratic equation with integer coefficients) with the distance from $a_n\alpha$ to the nearest integer tending to $0$, while $a_n$ grows slower than any prescribed unbounded multiple of $n$. Berend–Boshernitzan–Kolesnik–Kumar (2024) proved that every Hausdorff dimension $d\in[0,1]$ occurs as the dimension of the density-exceptional set of some integer sequence, and likewise for the natural subgroups $E_0(r)$, the set of $x$ with $r_n x\to 0$ mod $1$, and $E_f(r)$, the set of $x$ with the sum over $n$ of the distance from $r_n x$ to the nearest integer convergent; so the dimension spectrum of exceptional sets is fully realized between the sublacunary and lacunary extremes; for the uniform-distribution exceptional set itself, every dimension in $[0,1]$ already arises through Ruzsa's polynomial-growth family together with the lacunary sequences.

A quantitative companion problem concerns the almost-everywhere rate in Weyl's metric theorem: Baker (1981) proved $D_N((n_k x))=O((\log N)^{3/2+\varepsilon}/\sqrt{N})$ for almost every $x$ and every increasing integer sequence, while Berkes–Philipp (1994) showed that the exponent of the logarithmic term cannot be pushed below $1/2$; the 2023 survey of Aistleitner–Berkes–Tichy records that the precise rate 'is still open more than a hundred years later' and calls the optimal exponent 'one of the most fundamental problems in metric discrepancy theory'. For lacunary sequences the discrepancy obeys a law of the iterated logarithm — an almost-sure growth rate of order $\sqrt{N\log\log N}$, as for sums of independent random variables (Philipp 1975) — with the exact limiting constants for geometric sequences determined by Fukuyama (2008); the relevant greatest-common-divisor sums were analyzed by Aistleitner–Berkes–Seip (2015) and Lewko–Radziwiłł (2017), with the law of the iterated logarithm under intermediate gap conditions treated by Aistleitner (2010).

Problem Statement

Let $n_1<n_2<n_3<\cdots$ be a strictly increasing sequence of positive integers. For a real number $x$, the dilation $(n_k x)$ is said to be uniformly distributed modulo $1$ if the proportion of indices $k\le N$ with fractional part $\{n_k x\}$ in $[a,b)$ tends to $b-a$ for every interval $[a,b)\subseteq[0,1)$; equivalently, by Weyl's criterion, if $\frac{1}{N}\sum_{k\le N}e^{2\pi i h n_k x}\to 0$ for every nonzero integer $h$. Define the exceptional set

$E(n_\bullet)=\{\,x\in\mathbb{R} : (n_k x)_{k\ge1} \text{ is not uniformly distributed mod } 1\,\}.$

By Weyl's metric equidistribution theorem, $E(n_\bullet)$ is always a set of Lebesgue measure zero, and it always contains $\mathbb{Q}$, since for rational $x$ the fractional parts $\{n_k x\}$ take only finitely many values and hence are not uniformly distributed.

Task. Give, with proof, a characterization of the strictly increasing integer sequences for which the exceptional set is minimal — that is, the sequences with

$E(n_\bullet)=\mathbb{Q},$

equivalently the sequences such that $(n_k x)$ is uniformly distributed mod $1$ for every irrational $x$ — in terms of explicit structural properties of $(n_k)$: growth and gap conditions, multiplicative or arithmetical structure, or other combinatorial invariants, established as necessary and sufficient conditions.

A solution settles Weyl's uniform-distribution strengthening (1916, §7, p. 344) of the denseness question posed by Hardy and Littlewood (1914), together with the question Weyl left undecided (1916, p. 345) of whether the exceptional set of his metric theorem can contain numbers beyond the rationals.

References

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