Exponential digit complexity beyond the Bugeaud--Kim threshold

[Authors]

MSC 2020: 11A63, 11J82 (primary), 68R15 (secondary)

Keywords: subword complexity, Mahler function, Sturmian word, lacunary series, irrationality exponent

Abstract

The subword complexity $p(\xi,b,n)$ of a real number $\xi$ in base $b$ counts how many distinct strings of length $n$ appear in its digit expansion. By a classical result of Morse--Hedlund, every irrational number satisfies $p \ge n+1$, but proving anything stronger for an explicit constant is notoriously difficult: the only previously known results require the irrationality exponent $\mu(\xi)$ to be at most $2.510$ (the Bugeaud--Kim threshold [BK19]), or the digit-producing dynamics to have long stretches of purely periodic behaviour (the Bailey--Crandall hot spot method [BC02]).

We introduce an epoch-expansion technique that bypasses both barriers, and use it to prove that a broad family of lacunary sums --- constants of the form $\xi=\sum a_k/(c^{g(k)}b^{f(k)})$ with $\gcd(b,c)=1$ and rapidly increasing exponents $f$ --- have much richer digit structure than irrationality alone guarantees. Concretely, the number of distinct length-$n$ digit strings satisfies $p(\xi,b,n)-n\to+\infty$, with quantitative lower bounds that mirror the growth of the spacing $f$: quadratic spacing yields at least quadratically many strings, exponential spacing yields exponentially many, and so on.

Our strongest application concerns the twisted Mahler constants $M_{d,b,c}=\sum_{k\ge 0}1/(c^k b^{d^k})$. When $d\ge 3$ and $c>d^2$, these constants have irrationality exponent $\mu\ge d>2.510$ (so Bugeaud--Kim fails) and epoch-to-lattice ratio tending to zero (so hot spots fail), yet we obtain exponential complexity: $p(M_{d,b,c},b,n)\ge d^{n/\alpha-C}$ where $\alpha=\log_b c$. Further applications include partial theta function values, Tschakaloff function values, Fibonacci-exponent constants, and Rogers false theta function values. To our knowledge, these are the first complexity results for explicit constants beyond both known barriers.

1. Introduction

Motivation: measuring the complexity of digit expansions

Write a real number $\xi$ in an integer base $b\ge 2$:

$$\xi = 0.x_1 x_2 x_3 \cdots, \qquad x_n \in {0,1,\dots,b-1}.$$

One of the simplest questions one can ask about this expansion is: how many distinct patterns appear? The subword complexity function $p(\xi,b,n)$ counts the number of distinct blocks of $n$ consecutive digits $x_{i+1}x_{i+2}\cdots x_{i+n}$ that occur as $i$ ranges over all positions. For a rational number, the digit expansion is eventually periodic, so $p(n)$ is bounded. For an irrational number, a classical theorem of Morse and Hedlund [MH38] gives $p(n)\ge n+1$ for every $n$, and this bound is sharp: it is attained exactly by the Sturmian words, a family of aperiodic binary sequences that arise by coding an irrational rotation of the circle (see Lothaire [Lo02] for a comprehensive account). At the other extreme, a number is normal in base $b$ (in the sense of Borel [Bor09]) if every block of digits appears with the expected frequency $b^{-n}$; this forces $p(n)=b^n$, the maximum possible value. Almost every real number is normal in every base, but proving normality---or even any growth of $p(n)$ beyond $n+1$---for a specific constant such as $\pi$, $e$, or $\sqrt{2}$ is extraordinarily difficult.

What is known

There are essentially two prior techniques for proving that the digits of a given constant are more complex than the Morse--Hedlund minimum.

The Diophantine method. Ferenczi and Mauduit [FM97] first connected low subword complexity to transcendence: they showed that a number whose base-$b$ expansion satisfies $p(n) = n+1$ for all $n$ (a Sturmian number) is necessarily transcendental. Adamczewski and Bugeaud [AB07], using the deep Schmidt Subspace Theorem from Diophantine approximation, proved a much stronger result: the digit expansion of every algebraic irrational number $\xi$ satisfies $p(\xi,b,n)/n\to\infty$ for every base $b$. For transcendental constants, the key parameter governing what can be proved is the irrationality exponent

$$\mu(\xi) = \sup!\Bigl{\mu : |\xi - p/q| < q^{-\mu} \text{ for infinitely many } p/q\in\mathbb{Q}\Bigr},$$

which measures how well $\xi$ can be approximated by rationals. Bugeaud and Kim [BK19] showed that if $\mu(\xi) < \mu_0 := (25+4\sqrt{10})/15 \approx 2.510$, then $p(\xi,b,n) - n \to +\infty$. The reasoning is: a small irrationality exponent forces the digit expansion to avoid long near-repetitions; this is incompatible with the rigid self-similar structure of quasi-Sturmian words (sequences with $p(n)\le n+C$ for a constant $C$, characterized by Cassaigne [Ca97] as the image of a Sturmian word under a morphism); and the resulting structural contradiction yields complexity growth. The threshold $\mu_0=2.510\ldots$ is sharp: Bugeaud and Kim construct Sturmian numbers with $\mu = \mu_0$ and $p(n)=n+1$ for all $n$.

Prior to the present work, this method was the only known technique for proving $p(n)-n\to\infty$ for explicit transcendental constants. It covers Euler's number $e$ and its relatives $e^{1/m}$, $\tanh(1/m)$, and certain Bessel function quotients---all of which have $\mu=2$, well below the threshold.

The hot spot method. Bailey and Crandall [BC02] developed a completely different approach that proves a much stronger conclusion---full normality---for a special class of constants. The Stoneham numbers $\alpha_{b,c} = \sum_{k\ge 1} 1/(c^k b^{c^k})$, where $b$ and $c$ are coprime, have the property that successive nonzero terms are exponentially far apart. Between two consecutive terms, the digit-generating iteration $z_n = {b \cdot z_{n-1}}$ (fractional part of $b$ times the previous orbit point) evolves without perturbation, giving long "clean" stretches during which the orbit is algebraically controlled. Bailey and Crandall show that no short subinterval of $[0,1)$ is visited disproportionately often by this orbit (a "hot spot"), and this implies normality. The crucial structural requirement is that the epoch length (the number of unperturbed steps between consecutive terms of the series) is at least proportional to the lattice denominator of the orbit during that epoch.

The gap

Both methods leave large classes of constants untouched. The Diophantine method requires $\mu(\xi)<2.510$, yet for most transcendental constants the irrationality exponent is either unknown or too large: for $\pi$, the best bound is $\mu(\pi)\le 7.103$ (Zeilberger--Zudilin [ZZ20]), far above the threshold. The hot spot method requires the epoch-to-lattice ratio to remain bounded below by a positive constant, and this fails for series whose terms are more densely packed than in the Stoneham case.

Between these two regimes sits a large family of lacunary constants---sums whose nonzero terms are located at positions that grow faster than linearly but not exponentially (such as quadratic or polynomial positions)---for which no technique existed for proving anything about digit complexity beyond the trivial bound $p(n)\ge n+1$.

Our contribution: the epoch-expansion method

In this paper we introduce the epoch-expansion method, a technique for proving lower bounds on subword complexity that requires no information about the irrationality exponent and is orthogonal to the hot spot approach.

The key observation is elementary. Consider a constant of the form $\xi = \sum a_k / (c^{g(k)} b^{f(k)})$, where $b$ and $c$ are coprime and the exponents $f(k)$ grow with increasing gaps $f(k!+!1)-f(k) \to \infty$. The digit expansion of $\xi$ is naturally partitioned into epochs: long runs of digit positions between the locations $f(k)$ and $f(k!+!1)$ of successive terms. Inside each epoch, the orbit point $z_n = {b^n \xi}$---whose base-$b$ digits determine the digit expansion of $\xi$ from position $n$ onward---decomposes as a rational number with denominator $c^K$ plus a negligibly small tail. The coprimality $\gcd(b,c)=1$ places the numerator of this rational part on a $c$-adic lattice, and a key arithmetic observation controls how long two orbit points can produce identical digits: if positions $n$ and $n+s$ lie in the same epoch, they share at most $v_c(b^s-1)\cdot \log_b c + O(1)$ leading digits, where $v_c$ denotes the $c$-adic valuation. This bound grows only logarithmically in the epoch index, while the epochs themselves grow without bound. Eventually the epochs are long enough to contain more pairwise distinguishable starting positions than any quasi-Sturmian word can accommodate, and the complexity $p(n)$ is forced to exceed $n+C$ for every constant $C$.

Results

We present our results in order of increasing generality, developing the technique first for the partial theta values

$$\gamma_{b,c}:=\sum_{k\ge 1}\frac{1}{c^k b^{k^2}}, \qquad b\ge 2,\ c\ge 2,\ \gcd(b,c)=1,$$

which serve as the motivating case. These constants are values of a partial Jacobi theta function (related to the $q$-series $\Theta^+(q,z) = \sum_{k\ge 0} q^{k^2} z^k$); they escape both prior methods, since the epoch-to-lattice ratio tends to zero (defeating hot spots) and no bound on $\mu(\gamma_{b,c})$ is available (so the Diophantine method cannot be applied).

Theorem 1.1 (Qualitative complexity). For all coprime integers $b\ge 2$ and $c\ge 2$,

$$p(\gamma_{b,c},b,n)-n\longrightarrow +\infty \qquad (n\to\infty).$$

Theorem 1.2 (Quadratic lower bound). If $\alpha=\log_b c<2$, then there exists $n_0=n_0(b,c)$ such that

$$p(\gamma_{b,c},b,n)\ge \frac{(2-\alpha)^2}{16\alpha^2},n^2 \qquad\text{for all } n\ge n_0.$$

(We write $\alpha = \log_b c$ throughout the paper.) For a more striking application, we turn to the twisted Mahler constants. For integers $d\ge 2$, $b\ge 2$, $c\ge 2$ with $\gcd(b,c)=1$, define

$$M_{d,b,c} ;:=; \sum_{k\ge 0}\frac{1}{c^k,b^{,d^k}}.$$

These arise from the Mahler function $h(x) = \sum_{k\ge 0} x^{d^k}$, which satisfies $h(x) = x + h(x^d)$; see Nishioka [Ni96] for background on Mahler functions in transcendence theory. When $d\ge 3$ and $c > d^2$, these constants lie beyond the reach of both prior approaches simultaneously. On the Diophantine side, $\mu(M_{d,b,c}) = d$ (proved via the formal continued fraction of the associated Mahler function; see Badziahin [Bad19] and Rajchert [Raj24]), so $\mu \ge 3 > 2.510$ and the Bugeaud--Kim threshold is exceeded. On the hot spot side, Bailey--Crandall [BC02] prove normality of $M_{d,b,c}$ only when $d > \sqrt{c}$; when $c>d^2$ this condition fails. Nevertheless:

Theorem 1.3 ($\mu$-bypass). For all $d\ge 2$, $b\ge 2$, $c\ge 2$ with $\gcd(b,c)=1$,

$$p(M_{d,b,c},,b,,n)-n;\longrightarrow;+\infty \qquad(n\to\infty).$$

When $d < c$, the exponential growth of the epoch lengths $L_K = (d-1)d^K$ allows a much stronger, quantitative conclusion:

Theorem 1.4 (Exponential lower bound). For all $d\ge 2$, $b\ge 2$, $c\ge 2$ with $d<c$ and $\gcd(b,c)=1$, there exists a constant $C=C(d,b,c)$ such that

$$p(M_{d,b,c},,b,,n) ;\ge; d^{,n/\alpha,-,C}$$

for all sufficiently large $n$.

For comparison, the strongest quantitative result available from the Bugeaud--Kim method is $\limsup, p(n)/n\ge 4/3$ [BKK25], which applies only to constants satisfying a stringent irrationality exponent condition. Theorem 1.4 gives exponential growth in a regime where that method is provably inapplicable (see Remark 8.1).

All of the above are special cases of a general theorem. For integers $b\ge 2$, $c\ge 2$ with $\gcd(b,c)=1$, let $f\colon\mathbb{N}\to\mathbb{N}$ be strictly increasing with $L_K:=f(K!+!1)-f(K)\to\infty$, let $g\colon\mathbb{N}\to\mathbb{N}$ be strictly increasing with $g(K)\to\infty$ and $g(K)=o(f(K))$, and let $(a_k)_{k\ge 1}$ be integers with $\gcd(a_k,c)=1$ for all $k$. Assume further that

$$v_c(b^s-1)<g(K) \qquad\text{for all }1\le s\le L_K\text{ and all large }K. \tag{1}$$

(A sufficient condition is $\log L_K = o(g(K))$, since $v_c(b^s-1)=O(\log s)$. When $f$ grows exponentially, (1) can instead be verified via multiplicative order estimates; see Corollary 7.2 and (8).) Define

$$\xi;=;\sum_{k\ge 1}\frac{a_k}{c^{g(k)},b^{f(k)}}.$$

Theorem 1.5 (General epoch-expansion theorem). Under the hypotheses above, $p(\xi,b,n)-n\to+\infty$. Moreover, there exist constants $C_0=C_0(b,c)$ and $K_0=K_0(b,c,f,g)$ such that for all sufficiently large $n$,

$$p(\xi,b,n);\ge;\sum_{K=A_n}^{B_n}(L_K - n + 1),$$

where $A_n=\min{K\ge K_0: L_K\ge n}$ and $B_n=\max{K: g(K)\alpha + C_0 < n}$. When $B_n < A_n$ (which can happen for slowly growing $f$ with $\alpha \ge 2$), the qualitative conclusion $p(\xi,b,n)-n\to+\infty$ still holds via the Cassaigne argument of Theorem 1.1, applied with $g(K)$ in place of $K$.

The quantitative bound mirrors the spacing function: quadratic spacing ($f(k)=k^2$) gives $p(n)\ge c_0 n^2$; exponential spacing ($f(k)=d^k$) gives $p(n)\ge d^{n/\alpha-C}$. Theorems 1.1, 1.2, and 1.4 are corollaries of Theorem 1.5 (see Section 7); Theorem 1.3 requires an independent argument given in Section 8.

Proof architecture

We present the proofs for the partial theta values $\gamma_{b,c}$ independently of the general theorem, because they introduce the key ideas in a concrete setting. The architecture of the general proof is as follows.

All results rest on the epoch-expansion framework, whose applicability is determined by the structural properties of $\xi=\sum a_k/(c^{g(k)}b^{f(k)})$: coprime denominator ($\gcd(b,c)=1$, giving a $c$-adic lattice structure for the orbit); increasing gaps ($L_K \to \infty$, creating epochs of growing length); single-modulus lattice (all terms share the modulus $c$, making $v_c(b^s-1)$ the sole obstruction to digit agreement); and valuation control (condition (1)).

Given these properties, the proof operates in two modes. For the qualitative conclusion ($p(n)-n\to\infty$), we argue by contradiction: if $p(n)-n$ were bounded, Cassaigne's theorem [Ca97] would force the digit sequence to be quasi-Sturmian, producing long self-matches at certain gaps. We place such a matched pair inside a single large epoch and show that the within-epoch valuation bound limits the match length to $O(K\alpha)$, contradicting the Sturmian match length $\gg q_{k+1}$ since $K=O(\log q_{k+1})$. For the quantitative conclusion, we count distinct length-$n$ subwords directly: each epoch of length $L_K \ge n$ contributes $\sim L_K$ starting positions whose length-$n$ subwords are pairwise distinct (by the valuation bound within epochs, and by the lattice gap across epochs), and summing over eligible epochs yields the lower bound.

Further applications

Beyond partial theta values and twisted Mahler constants, Theorem 1.5 applies uniformly to every lacunary constant satisfying the hypotheses above:

(a) Tschakaloff function values $T_b(1/c)=\sum_{k\ge 1}1/(c^k b^{k(k+1)/2})$ ($f(k)=k(k+1)/2$, $L_K=K+1$; quadratic bound).

(b) Fibonacci-exponent constants $\sum_{k\ge 1}1/(c^k b^{F_k})$ ($f(k)=F_k$, $L_K=F_{K-1}$; exponential bound $p(n)\ge \varphi^{n/\alpha-C}$ when $\alpha$ is sufficiently small).

(c) Signed coefficients $a_k\in{\pm 1}$ ($\gcd(\pm 1,c)=1$ always); this covers values of Rogers false theta functions at rational arguments.

Organization

Section 2 collects notation and the Sturmian input. Section 3 gives the epoch decomposition of the orbit ${b^n\gamma_{b,c}}$. Section 4 proves the within-epoch and cross-epoch match bounds. Theorems 1.1 and 1.2 are proved in Sections 5 and 6. Section 7 states and proves the general epoch-expansion theorem (Theorem 1.5) and derives applications to Tschakaloff function values, Fibonacci-exponent constants, and signed-coefficient series. Section 8 treats twisted Mahler constants, proving Theorems 1.3 and 1.4; this includes the exponential bound in the $\mu$-bypass regime $d\ge 3$, $c>d^2$ where both prior methods are provably inapplicable. Section 9 discusses open questions.

2. Preliminaries

Throughout the paper, $b\ge 2$ and $c\ge 2$ are fixed coprime integers, and

$$\alpha=\log_b c>0.$$

2.1. Digits, orbit points, and complexity

For $x\in\mathbb{R}$, we write $\lfloor x \rfloor$ for the floor and ${x}=x-\lfloor x \rfloor$ for the fractional part.

We consider

$$\gamma_{b,c}=\sum_{k\ge 1}\frac{1}{c^k b^{k^2}},$$

and its orbit under multiplication by $b$:

$$z_n:={b^n\gamma_{b,c}},\qquad n\ge 0.$$

The base-$b$ digit sequence of $\gamma_{b,c}$ is then

$$x_n=\lfloor b z_{n-1} \rfloor,\qquad n\ge 1,$$

so that

$$\gamma_{b,c}=\sum_{n\ge 1} x_n b^{-n}.$$

Definition 2.1. For $n\ge 1$, the subword complexity of $\gamma_{b,c}$ in base $b$ is

$$p(\gamma_{b,c},b,n) = #\bigl{x_{i+1}x_{i+2}\cdots x_{i+n}: i\ge 0\bigr}.$$

When the parameters are clear, we abbreviate this to $p(n)$.

2.2. Match length

Definition 2.2. Let $\mathbf{w}=(w_n)_{n\ge 1}$ be an infinite word over a finite alphabet. For $n\ge 1$ and $g\ge 1$, the match length at position $n$ with gap $g$ is

$$\operatorname{Match}$${\mathbf{w}}(n,g) = \max\bigl{L\ge 0 : w{n+j}=w_{n+g+j}\ \text{for all }0\le j<L\bigr}.

For the digit sequence $\mathbf{x}=(x_n)${n\ge 1}x=(xn​)n≥1​ of $\gamma${b,c}, we write

$$M(n,g):=\operatorname{Match}_{\mathbf{x}}(n+1,g), \qquad n\ge 0,\ g\ge 1.$$

Equivalently, $M(n,g)$ is the largest $L\ge 0$ such that the first $L$ base-$b$ digits of $z_n$ and $z_{n+g}$ coincide.

2.3. Square epochs

For $K\ge 1$, define epoch $K$ to be the interval of positions

$$[K^2,(K+1)^2).$$

Its length is

$$L_K=(K+1)^2-K^2=2K+1.$$

If $n$ lies in epoch $K$, we write

$$e_n:=(K+1)^2-n, \qquad 1\le e_n\le 2K+1,$$

for the distance from $n$ to the next epoch boundary.

2.4. $c$-adic valuation

Let

$$c=p_1^{a_1}\cdots p_r^{a_r}$$

be the prime factorization of $c$. For a nonzero integer $m$, define

$$v_c(m):= \min_{1\le i\le r}\left\lfloor\frac{v_{p_i}(m)}{a_i}\right\rfloor,$$

so that $v_c(m)$ is the largest $k\ge 0$ such that $c^k\mid m$. We set $v_c(0)=+\infty$.

2.5. Cassaigne's theorem

A Sturmian word is an aperiodic binary word of minimal complexity, namely $p(n)=n+1$ for all $n\ge 1$. A word is quasi-Sturmian if its complexity is bounded above by $n+C$ for some constant $C$.

We use the following theorem of Cassaigne.

Theorem 2.3 (Cassaigne [Ca97]). For a non-eventually-periodic infinite word $\mathbf{u}$ over a finite alphabet, the following are equivalent:

(i) $\liminf_{n\to\infty}(p_{\mathbf{u}}(n)-n)<\infty$;

(ii) $\mathbf{u}$ is eventually quasi-Sturmian, i.e. there exist a finite word $W$, a Sturmian word $\mathbf{s}$ over ${0,1}$, and a morphism

$$\varphi:{0,1}\to \mathcal{A}^*$$

such that

$$\mathbf{u}=W,\varphi(\mathbf{s}).$$

2.6. Two Sturmian lemmas

The first lemma is the source of long local self-matches in a Sturmian word. Here $\operatorname{Match}_{\mathbf{s}}(n,g)$ denotes the match length of Section 2 applied to the Sturmian word $\mathbf{s}$.

Lemma 2.4 (Strong Sturmian match). Let $\mathbf{s}$ be a Sturmian word of slope $\beta$, and let $q_k$ be the denominators of the convergents of $\beta$. For all sufficiently large $k$, there exists

$$n_k\in [q_{k+1}/2,\ q_{k+1})$$

such that

$$\operatorname{Match}$${\mathbf{s}}(n_k+1,q_k)\ge \lfloor q{k+1}/16\rfloor.

Proof. Let $\varepsilon_k=|q_k\beta|$. By continued-fraction theory,

$$\frac{1}{q_{k+1}+q_k}<\varepsilon_k<\frac{1}{q_{k+1}},$$

hence $\varepsilon_k>1/(2q_{k+1})$ for large $k$.

A mismatch at shift $q_k$ among the first $m$ symbols can occur only if the starting point of the coding orbit falls within distance $\varepsilon_k$ of one of the $m$ discontinuity preimages. Thus the set of bad starting points is covered by at most $m$ intervals of total length at most $2m\varepsilon_k$.

Among the $q_{k+1}$ points ${j\beta}$, $0\le j<q_{k+1}$, the minimum spacing is at least $\varepsilon_k>1/(2q_{k+1})$. Therefore the number of bad starting points among these $q_{k+1}$ orbit points is at most

$$2m\varepsilon_k\cdot 2q_{k+1}+m\le 5m.$$

Take $m=\lfloor q_{k+1}/16 \rfloor$. Then at least

$$q_{k+1}-5m\ge \frac{11}{16}q_{k+1}$$

starting points have match length at least $m$. In particular, one such point lies in $[q_{k+1}/2,\ q_{k+1})$. ∎

The second lemma is a standard recurrence property of Sturmian words.

Lemma 2.5 (Recurrence of Sturmian factors). Let $\mathbf{s}$ be a Sturmian word with convergent denominators $q_k$. For $\ell \in [q_k, q_{k+1})$, the recurrence function satisfies

$$R(\ell) ;\le; q_{k+1} + q_k - 1.$$

In particular, every factor of length $\ell$ occurs in every block of length $q_{k+1} + q_k$.

Proof. See [Lo02, Ch. 2, Prop. 2.2.22]. ∎

3. Epoch decomposition and irrationality

We begin with the basic decomposition of $z_n$ inside a square epoch.

Lemma 3.1 (Epoch decomposition). Let $n$ lie in epoch $K$, so that $K^2\le n<(K+1)^2$. Then

$$z_n=\frac{P_n}{c^K}+T_n, \tag{2}$$

where:

(a) $P_n$ is an integer with $0\le P_n<c^K$ and

$$P_n\equiv b^{,n-K^2}\pmod c,$$

hence $\gcd(P_n,c)=1$;

(b)

$$T_n=\sum_{t\ge 1}\frac{1}{c^{K+t} b^{u_{n,t}}}, \qquad u_{n,t}:=(K+t)^2-n =e_n+(t-1)(2K+t+1).$$

In particular,

$$0<T_n\le \frac{16}{15},\frac{1}{c^{K+1}b^{e_n}}, \tag{3}$$

and

$$0<T_n-\frac{1}{c^{K+1}b^{e_n}} \le \frac{16}{15},\frac{1}{c^{K+2}b^{e_n+2K+3}}. \tag{4}$$

Proof. Write

$$b^n\gamma_{b,c} = \sum_{j=1}^{K}\frac{b^{n-j^2}}{c^j} + \sum_{j>K}\frac{b^{n-j^2}}{c^j}.$$

Set

$$Q_n:=\sum_{j=1}^{K} c^{K-j} b^{n-j^2}\in \mathbb{Z}.$$

Then

$$b^n\gamma_{b,c}=\frac{Q_n}{c^K}+T_n, \qquad T_n:=\sum_{j>K}\frac{b^{n-j^2}}{c^j}.$$

Let $P_n$ be the residue of $Q_n$ modulo $c^K$ in $[0,c^K)$. By (3) below and $e_n \ge 1$, $0 < T_n < 1/c^K$. Since $0 \le P_n < c^K$, we have $P_n/c^K + T_n < (c^K - 1)/c^K + 1/c^K = 1$. Taking fractional parts therefore yields

$$z_n=\frac{P_n}{c^K}+T_n.$$

Modulo $c$, every term in $Q_n$ with $j<K$ vanishes, while the term $j=K$ equals $b^{n-K^2}$. Thus

$$P_n\equiv Q_n\equiv b^{n-K^2}\pmod c,$$

and since $\gcd(b,c)=1$, we also have $\gcd(P_n,c)=1$.

For the tail, write $j=K+t$ with $t\ge 1$. Then

$$T_n = \sum_{t\ge 1}\frac{1}{c^{K+t} b^{(K+t)^2-n}} = \sum_{t\ge 1}\frac{1}{c^{K+t} b^{u_{n,t}}}.$$

Since

$$u_{n,t}=e_n+(t-1)(2K+t+1)\ge e_n+(t-1)(2K+3),$$

we get

$$T_n \le \frac{1}{c^{K+1}b^{e_n}} \sum_{t\ge 0}\left(\frac{1}{cb^{2K+3}}\right)^t.$$

As $cb^{2K+3}\ge 16$, the geometric series is bounded by $16/15$, which proves (3).

The same argument starting from $t=2$ gives

$$T_n-\frac{1}{c^{K+1}b^{e_n}} \le \frac{1}{c^{K+2}b^{e_n+2K+3}} \sum_{u\ge 0}\left(\frac{1}{cb^{2K+5}}\right)^u \le \frac{16}{15},\frac{1}{c^{K+2}b^{e_n+2K+3}}.$$

This proves (4). ∎

Lemma 3.2. The number $\gamma_{b,c}$ is irrational.

Proof. Let

$$S_K:=\sum_{k=1}^{K}\frac{1}{c^k b^{k^2}}.$$

Its denominator divides

$$D_K:=c^K b^{K^2}.$$

Moreover,

$$0<\gamma_{b,c}-S_K \le \sum_{j\ge K+1}\frac{1}{c^j b^{j^2}} \le \frac{2}{c^{K+1}b^{(K+1)^2}} = \frac{2}{D_{K+1}}$$

for $K$ large.

If $\gamma_{b,c}=p/q\in\mathbb{Q}$ and $S_K\ne p/q$, then

$$\frac{1}{qD_K}\le \left|\gamma_{b,c}-S_K\right|\le \frac{2}{D_{K+1}}.$$

Hence $D_{K+1}\le 2qD_K$, which is impossible for large $K$ because

$$\frac{D_{K+1}}{D_K}=c,b^{2K+1}\to\infty.$$

If instead $S_K=p/q$ for all large $K$, then $S_{K+1}>S_K$, again impossible. ∎

4. Match bounds

4.1. A basic comparison principle

If two numbers in $[0,1)$ have the same first $L$ base-$b$ digits, then they belong to the same $b$-adic interval of length $b^{-L}$. Therefore, whenever $z_n\ne z_{n+s}$,

$$M(n,s)\le \log_b!\bigl(1/|z_n-z_{n+s}|\bigr). \tag{5}$$

4.2. Within one epoch

Theorem 4.1 (Within-epoch match bound). There exist constants $K_{\mathrm{we}}=K_{\mathrm{we}}(b,c)$ and $C_{\mathrm{we}}=C_{\mathrm{we}}(b,c)$ such that the following holds.

Let $K\ge K_{\mathrm{we}}$, and let $n,n+s$ lie in the same epoch $K$ with $1\le s\le 2K+1$. Then

$$M(n,s)\le K\alpha + C_{\mathrm{we}}.$$

Proof. Let $e=e_n=(K+1)^2-n$. Since $n+s$ also lies in epoch $K$, we have

$$e_{n+s}=e-s\ge 1, \qquad\text{hence}\qquad e\ge s+1.$$

By Lemma 3.1,

$$z_n=\frac{P_n}{c^K}+T_n, \qquad z_{n+s}=\frac{P_{n+s}}{c^K}+T_{n+s},$$

with $0\le P_n,P_{n+s}<c^K$. Set

$$\Delta:=P_n-P_{n+s}.$$

Then

$$z_n-z_{n+s}=\frac{\Delta}{c^K}+(T_n-T_{n+s}). \tag{6}$$

To understand $\Delta$, define

$$Q_m:=\sum_{j=1}^{K} c^{K-j} b^{m-j^2}\in \mathbb{Z} \qquad (m=n,n+s).$$

Then $Q_m\equiv P_m\pmod{c^K}$, and

$$Q_{n+s}=b^sQ_n.$$

Hence

$$P_{n+s}\equiv b^s P_n\pmod{c^K},$$

so

$$\Delta\equiv (1-b^s)P_n\pmod{c^K}. \tag{7}$$

Set

$$\nu:=v_c(b^s-1).$$

Since $1\le s\le 2K+1$, one has $\nu=O(\log K)$. Indeed, if $p\mid c$ is a fixed prime factor of $c$ and $d_0=\operatorname{ord}_p(b)$, then the standard $p$-adic valuation formula $v_p(b^s-1)=v_p(b^{d_0}-1)+v_p(s/d_0)$ (when $d_0\mid s$; zero otherwise) gives

$$v_p(b^s-1)\le A_p+\log_p s$$

for a constant $A_p=A_p(b)$. As $c^\nu\mid (b^s-1)$ implies $p^\nu\mid (b^s-1)$, we get

$$\nu\le v_p(b^s-1)\le A_p+\log_p(2K+1)=O(\log K).$$

Thus $\nu<K$ for all $K\ge K_{\mathrm{we}}$.

Now (7) and $\gcd(P_n,c)=1$ imply

$$v_c(\Delta)=v_c(b^s-1)=\nu,$$

because divisibility by $c^r$ with $r<K$ is preserved under congruence modulo $c^K$. In particular, $\Delta\ne 0$, and therefore

$$\left|\frac{\Delta}{c^K}\right|\ge \frac{1}{c^{K-\nu}}. \tag{}$$

Next, since $n$ and $n+s$ lie in the same epoch,

$$T_{n+s}=b^s T_n.$$

Using Lemma 3.1 and $e\ge s+1$,

$$|T_n-T_{n+s}|=(b^s-1)T_n \le \frac{16}{15},\frac{b^s-1}{c^{K+1}b^e} \le \frac{16}{15},\frac{1}{c^{K+1}b}.$$

Combining this with (6) and (*),

$$|z_n-z_{n+s}| \ge \frac{1}{c^{K-\nu}}-\frac{16}{15},\frac{1}{c^{K+1}b} = \frac{1}{c^{K-\nu}} \left(1-\frac{16}{15c^{\nu+1}b}\right).$$

Since $b,c\ge 2$,

$$\frac{16}{15c^{\nu+1}b}\le \frac{16}{15\cdot 2\cdot 2}=\frac{4}{15}<\frac12.$$

Therefore

$$|z_n-z_{n+s}|\ge \frac{1}{2c^{K-\nu}}.$$

By (5),

$$M(n,s)\le \log_b(2c^{K-\nu})\le K\alpha+\log_b 2.$$

This proves the theorem. ∎

4.3. A local cross-epoch bound

The next proposition is the local estimate needed in the qualitative proof. The hypothesis on the epoch gap $\delta$ is the only place where the quantity

$$D(K_1):=\min!\left(K_1+2,\ \left\lfloor\frac{2K_1+2}{\alpha}\right\rfloor\right)$$

appears.

Proposition 4.2 (Local cross-epoch bound). There exist constants $K_{\mathrm{loc}}=K_{\mathrm{loc}}(b,c)$ and $C_{\mathrm{loc}}=C_{\mathrm{loc}}(b,c)$ such that the following holds.

Let $N$ lie in epoch $K_1$, let $N+s$ lie in a later epoch $K_2>K_1$, and set

$$\delta:=K_2-K_1,\qquad e_1:=(K_1+1)^2-N,\qquad e_2:=(K_2+1)^2-(N+s).$$

Assume

$$1\le s\le 3N \qquad\text{and}\qquad \delta\le D(K_1).$$

Then for all $K_1\ge K_{\mathrm{loc}}$,

$$M(N,s)\le (K_2+1)\alpha + \max(e_1,e_2)+C_{\mathrm{loc}}.$$

In particular,

$$M(N,s)=O(\sqrt N).$$

Proof. By Lemma 3.1,

$$z_N=\frac{P_1}{c^{K_1}}+T_1, \qquad z_{N+s}=\frac{P_2}{c^{K_2}}+T_2,$$

where $\gcd(P_1,c)=\gcd(P_2,c)=1$. Set

$$D:=c^\delta P_1-P_2.$$

Then $c\nmid D$ (because $c\mid c^\delta P_1$ but $c\nmid P_2$), hence $D\ne 0$, and

$$z_N-z_{N+s}=\frac{D}{c^{K_2}}+T_1-T_2.$$

Extract the first tail term at each position:

$$T_1=\frac{1}{c^{K_1+1}b^{e_1}}+R_1, \qquad T_2=\frac{1}{c^{K_2+1}b^{e_2}}+R_2.$$

By (4),

$$|R_1| \le \frac{16}{15},\frac{1}{c^{K_1+2}b^{e_1+2K_1+3}}, \qquad |R_2| \le \frac{16}{15},\frac{1}{c^{K_2+2}b^{e_2+2K_2+3}}. \tag{}$$

Thus

$$z_N-z_{N+s} = \frac{D}{c^{K_2}} +\frac{1}{c^{K_1+1}b^{e_1}} -\frac{1}{c^{K_2+1}b^{e_2}} +(R_1-R_2).$$

We split into three cases.

Case 1: $e_1\ge e_2$. Multiply by $c^{K_1+1}b^{e_1}$:

$$c^{K_1+1}b^{e_1}(z_N-z_{N+s}) = \frac{cDb^{e_1}+c^\delta-b^{e_1-e_2}}{c^\delta}+E.$$