Edit (June 2026). Since posting this question, the aggregate equidistribution statement of Q1 below has been formally verified in Lean 4 + Mathlib end-to-end. The complete proof — spectral gap, tensor factorisation, Fourier inversion, exponential discrepancy bound, and the cardinality lower bound $|\mathcal{A}_k(b)| \geq c \cdot \varphi(b)^k$ — is machine-checked with no sorry and no custom axioms. An #print axioms audit on the top-level theorem returns only the three standard Mathlib kernel axioms (propext, Classical.choice, Quot.sound). DOI: 10.5281/zenodo.20320999 (v2.0.0). This question therefore concerns only literature and naming, not correctness of the statement. I have rewritten the Questions section below to reflect the current state: Q1 is resolved, Q2 is partially resolved, and Q3 (single-family characterization) is the most interesting open question to me.

For a base $b \geq 2$ and a multiset $S$ of digits in $\{0, \dots, b-1\}$, let $$\mathcal{F}(S, b) = \{ n : \text{the base-} b \text{ digits of } n \text{ are a permutation of } S, \text{ no leading zero} \}.$$ Call $\mathcal{F}(S, b)$ a permutation family: it is the orbit of one integer under permutation of digit positions. Fix $|S| = k$ and write $\mathcal{F}_k(S, b)$.

I am interested in the distribution of $\mathcal{F}_k(S, b)$ modulo a prime $p \nmid b(b-1)$, and specifically in a cancellation phenomenon when one aggregates over $S$.

Single families need not equidistribute. Take $b = 10$, $p = 11$, so $\mathrm{ord}_{11}(10) = 2$ and $10^j \equiv (-1)^j \pmod{11}$. For $S = \{1^5, 3^5\}$ ($k = 10$), every $n \in \mathcal{F}_{10}(S, 10)$ satisfies $$n \equiv \sum_{j \text{ even}} d_j - \sum_{j \text{ odd}} d_j \pmod{11},$$ which takes only $6$ distinct values; an exact count gives residues $\{2, 9\}$ with probability $\approx 0.40$ each and five residues with probability $0$. The maximal bias $\max_a |\mathbb{P}(n \equiv a) - 1/11|$ stays $\approx 0.3$ for $k \in \{10, 12, 14, 16\}$ and does not decay. So pointwise equidistribution fails, unsurprisingly, because $\mathrm{ord}_p(b)$ is small and the multiset is poor in residues mod $p$.

The aggregate is essentially uniform. Now take the union over all admissible multisets (digits coprime to $10$, digital root coprime to $9$) at $k = 10$: there are $188$ such multisets, totalling $699{,}392$ integers, and the aggregate distribution mod $11$ has maximal bias $$\max_{a \in \mathbb{Z}/11\mathbb{Z}} |\mathbb{P}(n \equiv a) - 1/11| = 0.0039.$$ The deterministic per-family biases cancel almost exactly.

Q3 (main, open). Is there a clean Diophantine condition on the triple $(S, b, p)$ — phrased in terms of $\mathrm{ord}_p(b)$, the orbit $\{b^j \bmod p\}$, and the digit-multiplicity profile of $S$ — that characterizes failure of equidistribution of $\mathcal{F}_k(S, b)$ mod $p$? I.e., is there a statement of the form $$\mathcal{F}_k(S, b) \text{ equidistributes mod } p \text{ as } k \to \infty \iff [\text{explicit condition on } S, b, p]$$ in the literature?

Two natural angles I have considered but cannot locate in the standard references:

(a) Character-sum / Ramanujan-sum side. The per-family bias factors through sums of the form $\sum_{\sigma} e^{2\pi i t \cdot n(\sigma) / p}$ over the orbit of digit-position permutations. The orbit has length $k! / \prod_d m_d(S)!$ where $m_d(S)$ is the multiplicity of digit $d$ in $S$, and the bias is controlled by Ramanujan-type sums restricted to admissible digital roots. Is there a finite-group character-sum bound that captures the resonances arising specifically when $\mathrm{ord}_p(b)$ is small relative to $k$?

(b) Random-permutation / local-CLT side. Each $\mathcal{F}_k(S, b)$ is a single orbit under $\mathrm{Sym}(k)$ acting on digit positions. Local limit theorems for linear statistics of uniform random permutations (Fulman; Kammoun) give Gaussian fluctuations at scale $\sqrt{k}$ for linear statistics; my biases here are $O(1)$ and stable in $k$ along sequences $S = \{d_1^k, d_2^k\}$, so a different regime. Is there a "rigid" / "non-mixing" regime in this body of work that captures families where the permutation-orbit lies inside a proper sublattice of residues mod $p$?

Q1 and Q2 (resolved, kept for context). Q1 asked whether the aggregate ratio $\#\{n \in \bigcup_S \mathcal{F}_k(S, b) : n \equiv a \pmod p\} / \#\bigcup_S \mathcal{F}_k(S, b) \to 1/p$ with an effective rate. This is now established (machine-verified, see Edit at top): the limit is $1/p$ with explicit exponential rate $O(\gamma(b, p)^k)$, $\gamma(b, p) < 1$, by finite Fourier analysis on $\mathbb{Z}/p\mathbb{Z} \times \mathbb{Z}/(b-1)\mathbb{Z}$ combined with a spectral gap on the finite orbit $\{b^j \bmod p\}$. Q2 asked whether this reduces to Konyagin / Maynard–Nath / Mauduit–Rivat: the digital-root constraint becomes a congruence mod $b-1$ (Mauduit–Rivat regime), but the further congruence mod $p$ requires the orbit of $b$ mod $p$, which is not addressed by those papers. So those references are related but not directly applicable to the joint statement.

Any pointer to relevant literature — a paper, a survey, an MSC subclass, or even an opinion on whether a clean Diophantine characterization should exist — would be very helpful.

Recent Questions - MathOverflow

Message History

Delivery option(s)