Quadratic Digit-Energy Series Analysis

• Quadratic digit-energy series are analytic constructs based on squaring digit weights that extend the second moment concept to evaluate convergence in digital numeration systems.
• Their convergence, together with linear digit-energy series, is essential for establishing necessary and sufficient conditions for limit laws governing digital additive functions.
• These series underpin the infinite product structure of characteristic functions, enabling explicit error bounds and unifying limit theorems across various numeral expansion frameworks.

Quadratic digit-energy series are central analytic constructs that arise in the study of additive functions defined on digital expansions, playing an essential role in establishing limit theorems for value distributions of sequence-defined arithmetic functions. The quadratic digit-energy series generalizes the notion of the second moment in probability theory to the setting of $q$-additive, Cantor, Zeckendorf, and more generally, digitally additive functions in various numeration systems. Its convergence is both necessary and sufficient, along with the linear digit-energy series, for the empirical distribution of such functions to converge to a limiting distribution, thus providing a unifying analytic criterion across a vast family of digital expansions (Drmota et al., 2020, Verwee, 24 Dec 2025).

1. Definition and Formulation

Given a numeration base—typically arising from integer bases, linear recurrence relations, or generalized digital expansions—a function $f : \mathbb{N} \to \mathbb{R}$ is termed digitally additive (e.g., $q$-additive, $G$-additive, $Z$-additive) if it decomposes as a sum over the digits in the unique expansion of $n$ relative to the chosen numeration system. The quadratic digit-energy series is then defined as the sum of the squares of these digit weights across all digit positions and allowable digits:

• $q$-Expansion ($q$-additive functions):

$$S_2(f) = \sum_{j=0}^\infty\sum_{d=1}^{q-1} f(dq^j)^2$$

• General Linear Recurrence (G-additive functions):

$$\sum_{n=0}^{\infty} \;\sum_{1\le c\le \mathfrak{a}}\; f(c\,G_n)^2$$

where $G_n$ is the $n$th term in the recurrence sequence and $\mathfrak a$ is the maximal digit allowed at each position (Verwee, 24 Dec 2025).

• Zeckendorf (Fibonacci) Expansion (Z-additive functions):

$S_2^Z(f) = \sum_{k\ge2} f(F_k)^2$

with $F_k$ denoting the $k$th Fibonacci number (Drmota et al., 2020).

This series mirrors Kolmogorov’s three-series in classical probability, governing the variance accumulation and ensuring the tightness and continuity of the associated limiting distributions' characteristic function.

2. Digital Additive Functions and Energy Series

Digitally additive functions are determined by assigning weights to each permissible digit-position pair in a given numeration scheme. For a $q$-additive function, this reduces to assigning values $f(dq^j)$ for $d \in \{1, \dots, q-1\}$ and $j \ge 0$, so that for any integer $n$ with base-$q$ expansion without carry-overs,

$f(n) = \sum_{i=1}^r f(d_i q^{e_i}).$

The linear (first moment) and quadratic (second moment) digit-energy series are respectively:

$$S_1(f) = \sum_{j=0}^\infty\sum_{d=1}^{q-1} f(dq^j), \qquad S_2(f) = \sum_{j=0}^\infty\sum_{d=1}^{q-1} f(dq^j)^2$$

For more general expansions, e.g., Cantor or Ostrowski systems, the energy series are adapted with suitable normalizations depending on the expansion’s local base parameters or permitted digits (Drmota et al., 2020, Verwee, 24 Dec 2025).

3. Erdős–Wintner-Type Theorems and Necessary and Sufficient Criteria

The classical Erdős–Wintner theorem characterizes the empirical value distribution of additive functions via convergence of first and second moment series. Delange’s theorem established the direct analogue for $q$-additive functions: the empirical distribution function

$F_N(y) = \frac{1}{N} \#\{ n < N : f(n) \leq y \}$

converges pointwise if and only if both $S_1(f)$ and $S_2(f)$ converge (Drmota et al., 2020).

These two-series criteria generalize to:

• Cantor digital expansions [Theorem 4.1, (Drmota et al., 2020)]
• Zeckendorf (Fibonacci) expansions [Theorem 5.1, (Drmota et al., 2020)]
• General linear recurrences, Ostrowski, and Parry expansions (Verwee, 24 Dec 2025)

In all cases, the convergence of the quadratic digit-energy series captures the limiting variance contribution from local digit effects, and its divergence is the sole obstruction to a limit law.

4. Infinite Product Structure of Limiting Characteristic Functions

If both moment series converge, the limiting distribution’s characteristic function $\varphi(t)$ admits a canonical infinite-product structure:

• $q$-Expansion:

$$\varphi(t) = \prod_{j=0}^{\infty} \left( \frac{1}{q} \sum_{d=0}^{q-1} e^{it\,f(dq^j)} \right )$$

• Cantor Expansion:

$$\varphi(t) = \prod_{j=0}^{\infty} \left( \frac{1}{a_j} \sum_{d=0}^{a_j - 1} e^{it\,f(dq_j)} \right )$$

• Zeckendorf Expansion:

$$\varphi(t) = \frac{\sqrt{5}}{\gamma} \prod_{k=2}^{\infty} \frac{r_k(t)}{\gamma}$$

where $r_{k+1}(t) = 1 + e^{it f(F_k)} r_k(t)$ and $\gamma = \frac{1 + \sqrt{5}}{2}$

• General Linear Recurrence and Ostrowski Expansions (Verwee, 24 Dec 2025):

$$\Phi(t) = \frac{1}{\kappa} \prod_{k=1}^\infty \Phi_k(t), \quad \Phi_k(t) = \frac{r_k(t)}{\alpha}$$

with explicit forms in terms of digit-weights and local expansion parameters.

This infinite product structure confirms that the limiting law depends only locally on the digit system, and the "energy" contributed at each expansion place is encoded in the magnitude of the quadratic series’ summands.

5. Quantitative Effective Theorems and Error Estimates

Beyond existential statements, explicit quantitative bounds on the rate of convergence are derivable using the Berry–Esseen technique and Fourier analytic control from the infinite-product structure of the characteristic function:

• For $q$-additive functions and other expansion types, error estimates for $\|F_N - F\|_\infty$ are obtained involving $S_1(f)$, $S_2(f)$ and local accumulations of digit weights [Theorem 3.1, Corollary 3.2, (Drmota et al., 2020)]. For instance, if $f(2^j) \asymp j^{-\alpha}$ for $1<\alpha<2$, then

$\|F_N - F\|_\infty \ll (\log N)^{1-\alpha}$

• For Zeckendorf expansions, the error term includes factors of $\frac{\log N}{T}$ and the tail sum of $|f(F_k)|$ beyond a logarithmic cutoff [Theorem 5.2, (Drmota et al., 2020)].

Such explicit bounds elucidate both convergence rates and the role of slowly decaying digit weights.

6. Generalizations to Other Numeration Systems

Recent advances have established that the necessity and sufficiency of the quadratic digit-energy series' convergence, together with the linear one, extend robustly to expansions determined by:

• Linear recurrence bases under mild structural hypotheses (primitive companion matrix, Pisot–Vijayaraghavan root, uniqueness via Parry’s lex rule)
• Ostrowski numerations with bounded partial quotients
• Parry $\beta$-expansions for Pisot–Vijayaraghavan $\beta$ (Verwee, 24 Dec 2025)

In each setting, appropriate modifications to the digit set and weight assignments yield analogous quadratic energy series, and the same two-series criterion governs the emergence of limit laws and the analytic structure of characteristic functions.

Expansion type	Digit-energy series	Limiting characteristic function structure
$q$-additive	$S_2(f) = \sum_{j,d} f(dq^j)^2$	Infinite product: $\prod_j$ (local digit factors)
Zeckendorf (Fibonacci)	$S_2^Z(f) = \sum_{k\ge2} f(F_k)^2$	Recursive/infinite product for $\varphi(t)$
Linear recurrence	$\sum_{n,c} f(cG_n)^2$	Product over $k$: local factors $\Phi_k(t)=r_k(t)/\alpha$

7. Significance and Connections with Probability Theory

The convergence of the quadratic digit-energy series is the analogue of the convergence of the variance term in Kolmogorov’s three-series theorem. It is responsible for the tightness of the measures induced by $f(n)$ and for the continuity at zero of the limiting characteristic function, ensuring the validity of the central limit behavior and the existence of classical laws (e.g., Cantor–Lebesgue, uniform, or Gaussian laws under varying decay rates of the digit weights).

A plausible implication is that similar quadratic digit-energy criteria may also play a role in the study of more exotic numeration systems, as well as in the investigation of ergodic and probabilistic properties of digital sequences and their associated additive functions.

References: Delange (1972), Coquet (1983), Barat–Grabner (2004), Berkes–Gál–Pintz (1990), Tenenbaum–Verwee (2018) as discussed in (Drmota et al., 2020), and extensions to linear recurrent bases in (Verwee, 24 Dec 2025).