REVIEW
Periodic counting, carry synchronization, and primes in nonstandard numeration systems
Number Theory · Submitted 5 Nov 2026 · 11 references
Pith's verdict: Solid, well-proven paper delivering a complete asymptotic analysis for a broad class of nonstandard numeration systems, with genuinely new results on counting, Fourier expansions, synchronization, and prime distribution.
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Referee report
The paper
The paper claims
A single periodic profile Ψ_D governs the counting of representable integers and primes in nonstandard numeration systems, under explicit arithmetic conditions.
The reading
What carries the argument
The key machinery includes: (1) a digit recurrence $R_D(n) = \sum_{d\in D} R_D(\frac{n-d}{b})$ that leads to a monotone scaling limit $F_D(x) = \lim_{m\to\infty} q^{-m} A_D(b^m x)$ and defines $\Psi_D$; (2) an exact periodic factorization $F_D(t) = t^{-\alpha} C_D(t) H_D(\log_b t)$ of the Laplace generating function, which expresses $\Psi_D$’s Fourier coefficients as $\hat\Psi_D(k) = h_D(k)/\Gamma(1+\alpha+i\tau_k)$; (3) for complete residue systems, a finite carry automaton on $K_D = \{-M,\dots,0\}$ with transitions $h_a(c) = \lfloor (a+c - d_{a+c})/b \rfloor$ that determines membership in $S_D$; (4) an arithmetic synchronization criterion (Theorem 1.3) proved via a weighted averaging identity and a minimum principle on trigonometric polynomials, showing that $\gcd(D)=1$ is necessary and sufficient for the existence of a zero-synchronizing word; (5) an interval approximation that uses the synchronizing word to partition the representable set into determined cylinders (where membership is constant) up to an exponentially small exceptional set, and a transfer principle that converts any arithmetic weight with uniform interval cancellation into a weighted sum over $S_D$.
Extended reading notes
Core claim
Under the condition $\mu < q$, the summatory representation function $A_D(x)$ satisfies $A_D(x) = x^\alpha \Psi_D(\log_b x) + O_D(x^\beta)$ with a unique, strictly positive, H\"older continuous, $1$-periodic profile $\Psi_D$ (Theorem 1.1). The profile admits an explicit Fourier series $\Psi_D(u) = \sum_{k\in\mathbb{Z}} \frac{h_D(k)}{\Gamma(1+\alpha+i\tau_k)} e^{2\pi i k u}$ with $\tau_k = 2\pi k / \log b$, where $h_D(k)$ are the Fourier coefficients of a smooth auxiliary function $H_D$ arising from a periodic factorization of the Laplace generating function (Theorem 1.2). When $D$ is a complete residue system modulo $b$ containing zero, the carry automaton on $K_D = \{-M,\dots,0\}$ admits a zero-synchronizing word if and only if $\gcd(D)=1$ (Theorem 1.3). This synchronization yields an exponentially accurate interval approximation of $S_D$ and a transfer principle for arithmetic weights, leading to the prime number theorem $\pi_D(x) = \frac{x}{\log x} \Psi_D(\log_b x) + O_D\bigl(\frac{x}{(\log x)^2}\bigr)$ (Theorem 1.4) and an $L^1$ scaling limit $\|\mathbf{1}_{E_N} - \mathbf{1}_{E_D}\|_{L^1([0,1])} \leq C_D \rho_D^N$ (Theorem 1.5).
Load-bearing premise
For the general counting theorem, the essential condition is that the digit set D has strictly fewer digits than its maximum residue multiplicity (µ < q); for the prime number theorem and scaling limit, D must additionally be a complete residue system modulo b with gcd(D)=1.
Editorial extensions
If this is right
- The prime counting function $\pi_D(x)$ is asymptotically equivalent to $A_D(x)/\log x$, with the same periodic fluctuations as the integer counting function $A_D(x)$.
- The relative proportion $\pi_D(x)/\pi(x)$ equals $\Psi_D(\log_b x) + O_D(1/\log x)$, exhibiting persistent logarithmic oscillations when $\Psi_D$ is nonconstant.
- The Dirichlet series $\zeta_D(s) = \sum_{n\ge 1} R_D(n) n^{-s}$ extends meromorphically to $\mathbb{C}$ with candidate simple poles at $s = \alpha - j + i\tau_k$ ($j\in\mathbb{N}_0$, $k\in\mathbb{Z}$), and the residues are explicitly expressed in terms of $\hat\Psi_D(k)$ and Taylor coefficients of $C_D(t)$.
- The rescaled sets $E_N = \bigsqcup_{0\le n<b^N,\, n\in S_D} [n/b^N, (n+1)/b^N)$ converge in $L^1([0,1])$ to a measurable limit $E_D$ whose cumulative distribution is $\int_0^t \mathbf{1}_{E_D}(u)\,du = t\Psi_D(\log_b t)$.
- The transfer principle (Proposition 5.4) applies to any arithmetic function with uniform cancellation on intervals, such as the Möbius function $\mu(n)$, implying Möbius orthogonality for the indicator $\mathbf{1}_{S_D}$ in complete analogy with the Sarnak conjecture for synchronizing automatic sequences.
Reading between the lines
- The construction of a zero-synchronizing word is algorithmic; the minimal length $\ell(D)$ is an interesting combinatorial invariant that may be linked to the \v{C}ern\'{y} conjecture for synchronizing automata, and the paper explicitly invites quantitative bounds on $\ell(D)$ in terms of $b$ and $d^* = \max D$.
- The Fourier expansion of $\Psi_D$ can be computed numerically to high precision for any concrete $D$, enabling quantitative predictions about the distribution of representable numbers and primes; the rapid decay of $h_D(k)$ ensures that truncating the series at a modest $|k|$ yields accurate approximations.
- The interval approximation method does not depend on the specific arithmetic weight; it could be adapted to other digital constraints beyond complete residue systems, though the synchronization property leveraged here is particular to the complete-residue setting and would require new techniques for sparse digit sets like missing-digit sets.
- The error term in the prime number theorem for $S_D$ inherits the quality of the classical prime number theorem; improvements in the latter (e.g., under the Riemann Hypothesis) would directly sharpen the error in Theorem 1.4.
Original abstract
Let $b\ge2$ be an integer and let $D\subset\mathbb{N}_0$ be a finite digit set containing zero, with maximum residue multiplicity $\mu=\max_{0\le r<b}\#\{d\in D:d\equiv r\pmod b\}$ and cardinality $q=\#D\ge2$. Under the condition $\mu<q$, we prove that the summatory representation function satisfies an asymptotic formula with a strictly positive, H\"older continuous, $1$-periodic fluctuating profile and an error of $O_D(x^{\log_b\mu})$. An exact periodic factorization of the Laplace generating function furnishes an explicit, computable Fourier expansion for the periodic profile via Gamma factors, and yields the meromorphic continuation of the associated Dirichlet series to the entire complex plane. When $D$ is a complete residue system modulo $b$, we establish an arithmetic synchronization criterion showing that the associated finite carry system admits a zero-synchronizing word if and only if $\gcd(D)=1$. This synchronization yields an exponentially accurate interval approximation and a general transfer principle for arithmetic weights. Consequently, we derive a prime number theorem for representable primes governed by the same periodic profile, alongside an exponentially convergent measurable scaling limit of the representable sets.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
Let $b\ge2$ be an integer and let $D\subset\mathbb{N}_0$ be a finite digit set containing zero, with maximum residue multiplicity $\mu=\max_{0\le r<b}\#\{d\in D:d\equiv r\pmod b\}$ and cardinality $q=\#D\ge2$. Under the condition $\mu<q$, we prove that the summatory representation function satisfies an asymptotic formula with a strictly positive, H\"older continuous, $1$-periodic fluctuating profile and an error of $O_D(x^{\log_b\mu})$. An exact periodic factorization of the Laplace generating function furnishes an explicit, computable Fourier expansion for the periodic profile via Gamma factors, and yields the meromorphic continuation of the associated Dirichlet series to the entire complex plane. When $D$ is a complete residue system modulo $b$, we establish an arithmetic synchronization criterion showing that the associated finite carry system admits a zero-synchronizing word if and only if $\gcd(D)=1$. This synchronization yields an exponentially accurate interval approximation and a general transfer principle for arithmetic weights. Consequently, we derive a prime number theorem for representable primes governed by the same periodic profile, alongside an exponentially convergent measurable scaling limit of the representable sets.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Let $b\ge2$ be an integer and let $D\subset\mathbb{N}_0$ be a finite digit set containing zero, with maximum residue multiplicity $\mu=\max_{0\le r<b}\#\{d\in D:d\equiv r\pmod b\}$ and cardinality $q=\#D\ge2$. | background 0.98 | 0.13 | 0.02 | unclear 0.54 |
| s1 | Under the condition $\mu<q$, we prove that the summatory representation function satisfies an asymptotic formula with a strictly positive, H\"older continuous, $1$-periodic fluctuating profile and an error of $O_D(x^{\log_b\mu})$. | result 0.99 | 0.94 | 0.46 | asserted 1.00 |
| s2 | An exact periodic factorization of the Laplace generating function furnishes an explicit, computable Fourier expansion for the periodic profile via Gamma factors, and yields the meromorphic continuation of the associated Dirichlet series to the entire complex plane. | method 0.66 | 0.90 | 0.11 | asserted 1.00 |
| s3 | When $D$ is a complete residue system modulo $b$, we establish an arithmetic synchronization criterion showing that the associated finite carry system admits a zero-synchronizing word if and only if $\gcd(D)=1$. | result 0.96 | 0.91 | 0.09 | asserted 1.00 |
| s4 | This synchronization yields an exponentially accurate interval approximation and a general transfer principle for arithmetic weights. | result 0.91 | 0.90 | 0.09 | asserted 1.00 |
| s5 | Consequently, we derive a prime number theorem for representable primes governed by the same periodic profile, alongside an exponentially convergent measurable scaling limit of the representable sets. | result 0.98 | 0.94 | 0.47 | asserted 1.00 |
What was examined, and what was not
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References
Discussion
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