Borel Complexity of Normal Vectors

• Borel complexity computations are defined by sets that are Pi^0_3-complete, expressed as countable intersections of F_sigma sets with continuous reductions from any such set.
• The study employs recurrence sequences, Pisot bases, and symbolic dynamics to show that normality corresponds to uniform distribution mod 1 and is encoded in bi-infinite subshifts.
• These findings extend classical base‑b normal numbers to higher-dimensional systems, providing deep insights into descriptive set theory and the structure of recurrence relations.

A set in descriptive set theory is $\Pi^0_3$-complete if it can be expressed as a countable intersection of $F_\sigma$ (i.e., countable unions of closed sets) subsets, and if every other $\Pi^0_3$ set can be continuously reduced to it. The Borel complexity of sets of vectors normal for a fixed recurrence sequence provides deep insight into the structure of normality in extended numeration and dynamical regimes, connecting classical normal number sets with higher-dimensional recurrence behavior.

1. Normality in Recurrence Sequences and Companion Polynomials

For a recurrence sequence $x_n = \xi_1 \alpha_1^n + \xi_2 \alpha_2^n$ determined by the companion polynomial $P(X)$ (e.g., $P(X)=(X-\alpha_1)(X-\alpha_2)$), the notion of a vector $(\xi_1, \xi_2)$ being "normal" means that the sequence $(x_n)_{n \ge 0}$ is uniformly distributed modulo $1$: for every interval $[a, b) \subset [-1/2,1/2)$,

$$\lim_{N \to \infty} \frac{1}{N} \#\left\{ 0 \leq n < N \mid x_n \pmod 1 \in [a, b) \right\} = b-a.$$

The set of normal vectors for $P$ is $$N_P = \{ (\xi_1, \xi_2) \in \mathbb{R}^2 \mid (x_n) \text{ u.d. mod } 1 \}$$ (Kaneko et al., 27 Oct 2025). Under mild algebraic hypotheses (notably: no root on the unit circle), this set is Borel and specifically $\Pi^0_3$-complete.

2. Statement of Main Complexity Results

The principal theorem ((Kaneko et al., 27 Oct 2025), Theorem 1.1-1.2) states:

• For every $k \ge 0$ and a fixed recurrence polynomial $P(X)$ without roots on the unit circle, the set

$$N_{P,k} = \{ \mathbf{g} \in \Xi_k : (x_n(\mathbf{g})) \text{ is u.d. mod } 1 \}$$

is $\Pi^0_3$-complete in the Polish space $\Xi_k$ parameterizing initial values.

• Special case: If $|\alpha|$ is Pisot ($|\alpha| > 1$ and all conjugates $<1$ in modulus), then for

$$N_\alpha = \{ \xi \in \mathbb{R} : (\xi \alpha^n)_{n \ge 0} \text{ is u.d. mod } 1 \}$$

the set $N_\alpha$ is $\Pi^0_3$-complete, generalizing classical base-$b$ normal numbers (Ki–Linton (Airey et al., 2018)) to all Pisot bases (Kaneko et al., 27 Oct 2025).

3. Methodology: Symbolic Encoding and Subshifts

A central idea is encoding recurrence sequences' fractional parts by infinite words over a suitably chosen alphabet $\mathcal{A}$ (of size $2B+1$, with $B$ depending on $P$'s coefficients). Each sequence $(x_n)$ can be recovered from

$$s = (s_n)_{n \in \mathbb{Z}}, \quad s_n \in \mathcal{A},$$

via a convolution kernel explicitly defined in terms of $P$ (Kaneko et al., 27 Oct 2025), Propositions 2.1–2.2.

This symbolic correspondence realizes a bi-infinite subshift $\Sigma_P \subset \mathcal{A}^\mathbb{Z}$, encoding all recurrence sequences. Normality (u.d. mod 1) for a vector $\mathbf{g}$ translates to the genericity of the associated $s = \Psi(\mathbf{g})$ in $\Sigma_P$: every finite block $w$ must appear in $s$ with the proper "generic" frequency determined by Lebesgue measure.

The subshift $\Sigma_P$ arising for $P$ is proved to have the right feeble specification property, a minimal form of mixing sufficient to guarantee that the set of generic points is $\Pi^0_3$-complete (Airey et al., 2018, Kaneko et al., 27 Oct 2025).

4. Descriptive Complexity and Completeness Arguments

The $\Pi^0_3$ membership of $N_P$ is shown via cylinder-block frequency conditions:

• For each block $w$ and tolerance $\epsilon > 0$, the set of $s$ whose empirical frequency approaches the limit within $\epsilon$ is $F_\sigma$.
• "Normality" requires these block-frequency conditions hold for all $w$ and $\epsilon$; thus $N_P$ is a countable intersection of $F_\sigma$ sets, i.e., $\Pi^0_3$.

$\Pi^0_3$-hardness follows from a Wadge reduction:

• The standard $\Pi^0_3$-complete set $$C_3 = \{ \beta \in \mathbb{N}^\mathbb{N} : \liminf \beta(n) \to \infty \}$$ in the Baire space.
• A continuous coding $F : \mathbb{N}^{\mathbb{N}} \to \Xi_k$ is constructed so that $F(\beta)$ is normal (in the sense above) if and only if $\beta(n) \to \infty$.
• The construction uses concatenations of "good" and "zero" blocks whose lengths depend on $\beta(n)$; long "good" blocks drive the sequence towards genericity, while bounded $\beta(n)$ prevent genericity (Kaneko et al., 27 Oct 2025), Lemmas 4.1–4.3.

5. Relation to Classical Normal Numbers and Broader Impact

This work subsumes the Ki–Linton (Airey et al., 2018) result that the set of base-$b$ normal numbers is $\Pi^0_3$-complete and extends it to:

• Numbers normal in any Pisot base,
• Vectors normal for arbitrary recurrence relations.

For general numeration systems or continued fractions, symbolic codings yield subshifts of specification type, ensuring $\Pi^0_3$-completeness of the normal set (Airey et al., 2018).

The methodology applies broadly: any system where normality corresponds to genericity for an invariant measure on a subshift with right feeble specification property will have the normal set $\Pi^0_3$-complete.

6. Open Problems and Further Directions

Areas for further research include:

• Extending results to non-Pisot bases (where digit expansions lack specification),
• Joint normality for multiple recurrence sequences, involving higher complexity classes such as $D_2(\Pi^0_3)$,
• Finer descriptive complexity above $\Pi^0_3$ for systems lacking strong mixing or with more irregular limiting behavior.

7. Table: Main $\Pi^0_3$ Normal Sets by Setting

System	Object(s)	Completeness Level	Reference
Base-$b$ normal	$\{\xi \mid (\xi b^n) \text{ u.d. mod } 1\}$	$\Pi^0_3$-complete	(Airey et al., 2018)
Pisot normal	$\{\xi \mid (\xi \alpha^n) \text{ u.d. mod } 1\}$ ($|\alpha|$ Pisot)	$\Pi^0_3$-complete	(Kaneko et al., 27 Oct 2025)
Recurrence normal	$$\{ (\xi_1, \xi_2) \mid x_n = \xi_1 \alpha_1^n + \xi_2 \alpha_2^n \text{ u.d. mod } 1 \}$$	$\Pi^0_3$-complete	(Kaneko et al., 27 Oct 2025)

• H. Kaneko, B. Mance, "Borel Complexity of the set of vectors normal for a fixed recurrence sequence" (Kaneko et al., 27 Oct 2025)
• D. Airey, S. Jackson, D. Kwietniak, B. Mance, "Borel complexity of sets of normal numbers via generic points in subshifts with specification" (Airey et al., 2018)
• H. Ki, T. Linton, "Normal numbers and subsets of $\mathbb{N}$ with given densities," Fund. Math. 144 (1994), 163–179.

In summary, for broad classes of recurrence and numeration systems—including all Pisot bases—the set of normal vectors or numbers is precisely $\Pi^0_3$-complete in the Borel hierarchy, as established by Kaneko–Mance (Kaneko et al., 27 Oct 2025) and related works. This provides a robust uniform answer for normality in analytic and dynamical settings, marking a boundary in the complexity of natural number-theoretic properties.