---
title: Borel Complexity of Normal Vectors
url: https://www.emergentmind.com/topics/borel-complexity-computations
type: topic
---

# Borel Complexity of Normal Vectors

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 $N_P = \{ (\xi_1, \xi_2) \in \mathbb{R}^2 \mid (x_n) \text{ u.d. mod } 1 \}$ [2510.23380]. 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 ([2510.23380], 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 [1811.04450]) to all Pisot bases [2510.23380].

## 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 $P$ [2510.23380], 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 [1811.04450], [2510.23380].

## 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 [2510.23380], Lemmas 4.1–4.3.

## 5. Relation to Classical Normal Numbers and Broader Impact

This work subsumes the Ki–Linton [1811.04450] 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 [1811.04450].

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 | [1811.04450] |
| Pisot normal     | $\{\xi \mid (\xi \alpha^n) \text{ u.d. mod } 1\}$ ($|\alpha|$ Pisot) | $\Pi^0_3$-complete | [2510.23380] |
| 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 | [2510.23380] |

## References

- H. Kaneko, B. Mance, "Borel Complexity of the set of vectors normal for a fixed recurrence sequence" [2510.23380]
- D. Airey, S. Jackson, D. Kwietniak, B. Mance, "Borel complexity of sets of normal numbers via generic points in subshifts with specification" [1811.04450]
- 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 [2510.23380] 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.

Source: https://www.emergentmind.com/topics/borel-complexity-computations