Beatty Multiple Shift Overview

• Beatty multiple shift is a symbolic system defined by constraints along two Beatty index sets, characterized by real parameters and their arithmetic overlaps.
• It extends multiplicative and affine shifts by employing density sequences from Beatty overlaps to compute explicit Hausdorff and Minkowski dimension formulas.
• Key insights include the pivotal role of the density sequence in recovering classical results and posing open challenges in determining the full density structure in region ⟨6⟩.

A Beatty multiple shift is a symbolic dynamical system in which adjacency constraints are imposed along two Beatty index sets rather than along arithmetic progressions. In the formulation introduced in "Hausdorff dimensions of Beatty multiple shifts" (Ban et al., 15 Jul 2025), one fixes an alphabet $\mathcal A=\{0,1,\dots,m-1\}$ with $m\ge 2$, an $m\times m$ binary matrix $A$, and real parameters $1\le \alpha<\gamma$ and $\beta,\delta\in\mathbb R$, and defines

$$X_A^{[\alpha,\beta,\gamma,\delta]} = \left\{ x=(x_i)_{i=1}^\infty\in \mathcal A^{\mathbb N} : A\!\left(x_{\lfloor \alpha k+\beta\rfloor},\,x_{\lfloor \gamma k+\delta\rfloor}\right)=1 \text{ for all } k\in\mathbb N \right\}.$$

This construction contains earlier multiplicative and affine multiple shifts as special cases, but its dimension theory is controlled by the arithmetic overlap of two Beatty sequences. The decisive input is a density sequence extracted from that overlap structure, and the resulting formulas link Hausdorff and Minkowski dimensions to classical questions on Beatty coverings of the positive integers (Ban et al., 15 Jul 2025).

1. Definition and antecedents

The immediate antecedent of the Beatty multiple shift is the multiplicative shift of finite type

$$X_A^{(q)} = \left\{ x=(x_i)_{i=1}^\infty\in \mathcal A^{\mathbb N}: A(x_k,x_{qk})=1 \text{ for all } k\in\mathbb N \right\},$$

followed by the affine multiple SFT

$$X_A^{\langle p,a,q,b\rangle} = \left\{ x=(x_i)_{i=1}^\infty\in \mathcal A^{\mathbb N}: A(x_{pk+a},x_{qk+b})=1 \text{ for all } k\in\mathbb N \right\}.$$

The Beatty model replaces the integer-affine indices $pk+a$ and $m\ge 2$0 by the Beatty expressions $m\ge 2$1 and $m\ge 2$2, thereby extending the affine theory from integer parameters to real parameters (Ban et al., 15 Jul 2025).

This generalization is exact: if $m\ge 2$3, then

$m\ge 2$4

Accordingly, the Beatty multiple shift contains the affine multiple SFT as a special case, and the classical multiplicative shift is recovered from the affine model by taking $m\ge 2$5 and $m\ge 2$6 (Ban et al., 15 Jul 2025).

The terminology combines two established traditions. In symbolic dynamics, “multiple shift” refers to constraints imposed simultaneously along more than one index map. In number theory, Beatty sequences and shifted Beatty sequences are classical objects. The new feature is that the two traditions are fused into a single object whose geometry depends on the distribution of

$m\ge 2$7

2. Arithmetic decomposition by Beatty overlap

The arithmetic core of the theory is the pair of sets

$m\ge 2$8

The paper defines a map

$m\ge 2$9

which is well-defined because different $m\times m$0 cannot produce the same $m\times m$1 when $m\times m$2 (Ban et al., 15 Jul 2025).

The positive integers are then decomposed into classes $m\times m$3, indexed by $m\times m$4, according to how the orbit under $m\times m$5 interacts with the overlap of the two Beatty sets. The description given in (Ban et al., 15 Jul 2025) is as follows. $m\times m$6 consists of integers in neither sequence. $m\times m$7 consists of points that move from $m\times m$8 to $m\times m$9 in one step. For $A$0, $A$1 contains points that remain in the intersection for $A$2 steps and then exit. The set $A$3 contains points that remain forever in the intersection under iteration.

For each $A$4, let $A$5 be the asymptotic density of $A$6. The sequence

$A$7

is the decisive arithmetic input in the dimension formulas. The paper emphasizes that this decomposition is closely related to the classical disjoint covering of the positive integers by Beatty sequences: the densities record how the two Beatty sets overlap, miss, or eventually synchronize (Ban et al., 15 Jul 2025).

This arithmetic viewpoint is the main structural departure from earlier affine models. In the integer-affine setting, the relevant combinatorics is controlled by congruence classes and integer scaling. In the Beatty setting, the coefficients of the dimension formulas are determined by densities arising from the overlap structure of two non-homogeneous Beatty sequences.

3. Hausdorff and Minkowski dimension theory

The principal theorem of (Ban et al., 15 Jul 2025) gives explicit dimension formulas in terms of the density sequence $A$8. Assume that $A$9 is an $1\le \alpha<\gamma$0 irreducible binary matrix, that $1\le \alpha<\gamma$1, that $1\le \alpha<\gamma$2, and that all densities $1\le \alpha<\gamma$3 exist. Under these hypotheses, the Minkowski dimension of $1\le \alpha<\gamma$4 is given by an explicit series involving $1\le \alpha<\gamma$5, the ratio $1\le \alpha<\gamma$6, and the quantities $1\le \alpha<\gamma$7. If $1\le \alpha<\gamma$8 is primitive, the Hausdorff dimension is also given explicitly, through a formula involving $1\le \alpha<\gamma$9 and a unique positive vector $\beta,\delta\in\mathbb R$0 satisfying

$\beta,\delta\in\mathbb R$1

As stated in the paper, this is the generalized eigenvector relation appearing in the thermodynamic formalism for these shifts (Ban et al., 15 Jul 2025).

The interpretation of the terms is explicit. The coefficients $\beta,\delta\in\mathbb R$2 measure how much of the integer set falls into each Beatty dynamical regime. The factor $\beta,\delta\in\mathbb R$3 reflects the geometric growth of the index map $\beta,\delta\in\mathbb R$4, since $\beta,\delta\in\mathbb R$5 expands roughly by $\beta,\delta\in\mathbb R$6. The quantities $\beta,\delta\in\mathbb R$7 count admissible words of length $\beta,\delta\in\mathbb R$8 along a length-$\beta,\delta\in\mathbb R$9 constraint chain. In the Hausdorff formula, $$X_A^{[\alpha,\beta,\gamma,\delta]} = \left\{ x=(x_i)_{i=1}^\infty\in \mathcal A^{\mathbb N} : A\!\left(x_{\lfloor \alpha k+\beta\rfloor},\,x_{\lfloor \gamma k+\delta\rfloor}\right)=1 \text{ for all } k\in\mathbb N \right\}.$$0 corresponds to indices with no constraint propagation, the coefficients $$X_A^{[\alpha,\beta,\gamma,\delta]} = \left\{ x=(x_i)_{i=1}^\infty\in \mathcal A^{\mathbb N} : A\!\left(x_{\lfloor \alpha k+\beta\rfloor},\,x_{\lfloor \gamma k+\delta\rfloor}\right)=1 \text{ for all } k\in\mathbb N \right\}.$$1 for $$X_A^{[\alpha,\beta,\gamma,\delta]} = \left\{ x=(x_i)_{i=1}^\infty\in \mathcal A^{\mathbb N} : A\!\left(x_{\lfloor \alpha k+\beta\rfloor},\,x_{\lfloor \gamma k+\delta\rfloor}\right)=1 \text{ for all } k\in\mathbb N \right\}.$$2 correspond to finite propagation depth, and $$X_A^{[\alpha,\beta,\gamma,\delta]} = \left\{ x=(x_i)_{i=1}^\infty\in \mathcal A^{\mathbb N} : A\!\left(x_{\lfloor \alpha k+\beta\rfloor},\,x_{\lfloor \gamma k+\delta\rfloor}\right)=1 \text{ for all } k\in\mathbb N \right\}.$$3 measures the contribution of infinite propagation (Ban et al., 15 Jul 2025).

A further structural criterion is inherited from earlier multiple-shift theories: Hausdorff and Minkowski dimensions coincide if and only if the matrix $$X_A^{[\alpha,\beta,\gamma,\delta]} = \left\{ x=(x_i)_{i=1}^\infty\in \mathcal A^{\mathbb N} : A\!\left(x_{\lfloor \alpha k+\beta\rfloor},\,x_{\lfloor \gamma k+\delta\rfloor}\right)=1 \text{ for all } k\in\mathbb N \right\}.$$4 has equal row sums (Ban et al., 15 Jul 2025). In this sense, uniform local combinatorics of the constraint matrix remains the exact condition for equality of the two dimensions, even though the ambient index geometry has become Beatty rather than affine.

4. Explicit density regimes and recovery of affine results

The arithmetic theorem of (Ban et al., 15 Jul 2025) divides the parameter space into regions $$X_A^{[\alpha,\beta,\gamma,\delta]} = \left\{ x=(x_i)_{i=1}^\infty\in \mathcal A^{\mathbb N} : A\!\left(x_{\lfloor \alpha k+\beta\rfloor},\,x_{\lfloor \gamma k+\delta\rfloor}\right)=1 \text{ for all } k\in\mathbb N \right\}.$$5, and in several of these regions the density sequence $$X_A^{[\alpha,\beta,\gamma,\delta]} = \left\{ x=(x_i)_{i=1}^\infty\in \mathcal A^{\mathbb N} : A\!\left(x_{\lfloor \alpha k+\beta\rfloor},\,x_{\lfloor \gamma k+\delta\rfloor}\right)=1 \text{ for all } k\in\mathbb N \right\}.$$6 is computed explicitly.

Region	Density sequence $$X_A^{[\alpha,\beta,\gamma,\delta]} = \left\{ x=(x_i)_{i=1}^\infty\in \mathcal A^{\mathbb N} : A\!\left(x_{\lfloor \alpha k+\beta\rfloor},\,x_{\lfloor \gamma k+\delta\rfloor}\right)=1 \text{ for all } k\in\mathbb N \right\}.$$7	Status
$$X_A^{[\alpha,\beta,\gamma,\delta]} = \left\{ x=(x_i)_{i=1}^\infty\in \mathcal A^{\mathbb N} : A\!\left(x_{\lfloor \alpha k+\beta\rfloor},\,x_{\lfloor \gamma k+\delta\rfloor}\right)=1 \text{ for all } k\in\mathbb N \right\}.$$8	$$X_A^{[\alpha,\beta,\gamma,\delta]} = \left\{ x=(x_i)_{i=1}^\infty\in \mathcal A^{\mathbb N} : A\!\left(x_{\lfloor \alpha k+\beta\rfloor},\,x_{\lfloor \gamma k+\delta\rfloor}\right)=1 \text{ for all } k\in\mathbb N \right\}.$$9	explicit
$$X_A^{(q)} = \left\{ x=(x_i)_{i=1}^\infty\in \mathcal A^{\mathbb N}: A(x_k,x_{qk})=1 \text{ for all } k\in\mathbb N \right\},$$0	$$X_A^{(q)} = \left\{ x=(x_i)_{i=1}^\infty\in \mathcal A^{\mathbb N}: A(x_k,x_{qk})=1 \text{ for all } k\in\mathbb N \right\},$$1	explicit
$$X_A^{(q)} = \left\{ x=(x_i)_{i=1}^\infty\in \mathcal A^{\mathbb N}: A(x_k,x_{qk})=1 \text{ for all } k\in\mathbb N \right\},$$2	$$X_A^{(q)} = \left\{ x=(x_i)_{i=1}^\infty\in \mathcal A^{\mathbb N}: A(x_k,x_{qk})=1 \text{ for all } k\in\mathbb N \right\},$$3	explicit
$$X_A^{(q)} = \left\{ x=(x_i)_{i=1}^\infty\in \mathcal A^{\mathbb N}: A(x_k,x_{qk})=1 \text{ for all } k\in\mathbb N \right\},$$4	separate arithmetic description in the appendix	partial explicitness
$$X_A^{(q)} = \left\{ x=(x_i)_{i=1}^\infty\in \mathcal A^{\mathbb N}: A(x_k,x_{qk})=1 \text{ for all } k\in\mathbb N \right\},$$5	determine $$X_A^{(q)} = \left\{ x=(x_i)_{i=1}^\infty\in \mathcal A^{\mathbb N}: A(x_k,x_{qk})=1 \text{ for all } k\in\mathbb N \right\},$$6	open

Region $$X_A^{(q)} = \left\{ x=(x_i)_{i=1}^\infty\in \mathcal A^{\mathbb N}: A(x_k,x_{qk})=1 \text{ for all } k\in\mathbb N \right\},$$7 is singled out by an explicit open problem: determine the full sequence $$X_A^{(q)} = \left\{ x=(x_i)_{i=1}^\infty\in \mathcal A^{\mathbb N}: A(x_k,x_{qk})=1 \text{ for all } k\in\mathbb N \right\},$$8 for parameters in that region (Ban et al., 15 Jul 2025). This is the main unresolved arithmetic input in the current dimension theory.

When the parameters are specialized to integers, the Beatty formulas recover the earlier affine theory. The paper splits the integer parameter space into four regions $$X_A^{(q)} = \left\{ x=(x_i)_{i=1}^\infty\in \mathcal A^{\mathbb N}: A(x_k,x_{qk})=1 \text{ for all } k\in\mathbb N \right\},$$9, states the corresponding density formulas, and recovers Theorem 3 and Theorem 4 of Ban–Hu–Lai–Liao on Hausdorff and Minkowski dimensions of affine multiple SFTs (Ban et al., 15 Jul 2025). Thus the Beatty model is not merely analogous to the affine model; it strictly contains it and reproduces its established results.

This recovery statement clarifies the role of Beatty parameters. Real-valued $$X_A^{\langle p,a,q,b\rangle} = \left\{ x=(x_i)_{i=1}^\infty\in \mathcal A^{\mathbb N}: A(x_{pk+a},x_{qk+b})=1 \text{ for all } k\in\mathbb N \right\}.$$0 do not simply perturb an existing integer theory; they generate a broader arithmetic classification whose integer points collapse back to the known affine formulas.

5. Shifted Beatty sequences in related literatures

The symbolic-dynamical Beatty multiple shift should be distinguished from earlier uses of shifted Beatty sequences in number theory and combinatorics. A standard shifted Beatty sequence is

$$X_A^{\langle p,a,q,b\rangle} = \left\{ x=(x_i)_{i=1}^\infty\in \mathcal A^{\mathbb N}: A(x_{pk+a},x_{qk+b})=1 \text{ for all } k\in\mathbb N \right\}.$$1

and the shift parameter $$X_A^{\langle p,a,q,b\rangle} = \left\{ x=(x_i)_{i=1}^\infty\in \mathcal A^{\mathbb N}: A(x_{pk+a},x_{qk+b})=1 \text{ for all } k\in\mathbb N \right\}.$$2 has long been used to study complementarity and partition phenomena. In the stadium model of "A Model for Pairs of Beatty Sequences" (Ginosar et al., 2011), the shifts encode starting positions of two athletes. For irrational $$X_A^{\langle p,a,q,b\rangle} = \left\{ x=(x_i)_{i=1}^\infty\in \mathcal A^{\mathbb N}: A(x_{pk+a},x_{qk+b})=1 \text{ for all } k\in\mathbb N \right\}.$$3 with

$$X_A^{\langle p,a,q,b\rangle} = \left\{ x=(x_i)_{i=1}^\infty\in \mathcal A^{\mathbb N}: A(x_{pk+a},x_{qk+b})=1 \text{ for all } k\in\mathbb N \right\}.$$4

Skolem’s theorem is recalled in the form that $$X_A^{\langle p,a,q,b\rangle} = \left\{ x=(x_i)_{i=1}^\infty\in \mathcal A^{\mathbb N}: A(x_{pk+a},x_{qk+b})=1 \text{ for all } k\in\mathbb N \right\}.$$5 and $$X_A^{\langle p,a,q,b\rangle} = \left\{ x=(x_i)_{i=1}^\infty\in \mathcal A^{\mathbb N}: A(x_{pk+a},x_{qk+b})=1 \text{ for all } k\in\mathbb N \right\}.$$6 eventually partition $$X_A^{\langle p,a,q,b\rangle} = \left\{ x=(x_i)_{i=1}^\infty\in \mathcal A^{\mathbb N}: A(x_{pk+a},x_{qk+b})=1 \text{ for all } k\in\mathbb N \right\}.$$7 if and only if

$$X_A^{\langle p,a,q,b\rangle} = \left\{ x=(x_i)_{i=1}^\infty\in \mathcal A^{\mathbb N}: A(x_{pk+a},x_{qk+b})=1 \text{ for all } k\in\mathbb N \right\}.$$8

That literature uses “shift” in the non-homogeneous Beatty sense rather than in the symbolic-dynamical sense (Ginosar et al., 2011).

A second strand concerns multiple shifted Beatty sequences in analytic number theory. In "Multivariate and quantitative Erdős-Kac laws for Beatty sequences" (Yip, 4 Feb 2026), the objects are

$$X_A^{\langle p,a,q,b\rangle} = \left\{ x=(x_i)_{i=1}^\infty\in \mathcal A^{\mathbb N}: A(x_{pk+a},x_{qk+b})=1 \text{ for all } k\in\mathbb N \right\}.$$9

and the main theorem states that for $pk+a$0, if $pk+a$1 for all $pk+a$2, then the centered and normalized vector converges in distribution to the standard $pk+a$3-dimensional Gaussian. The same paper proves that for any real $pk+a$4 and $pk+a$5, the Kolmogorov distance for a single Beatty sequence is bounded by

$pk+a$6

while no parameter-independent quantitative rate exists in general for joint convergence across multiple Beatty sequences (Yip, 4 Feb 2026).

A third strand studies intersections of multiple Beatty sequences with prime-rich sets. "Piatetski-Shapiro primes in the intersection of multiple Beatty sequences" (Guo et al., 2021) proves that if $pk+a$7 are irrational numbers of finite type and $pk+a$8 are linearly independent over $pk+a$9, then for $m\ge 2$00 there exist infinitely many primes in

$m\ge 2$01

with asymptotic main term

$m\ge 2$02

up to an error $m\ge 2$03 (Guo et al., 2021). Here again, “multiple” refers to simultaneous Beatty constraints, not to a symbolic shift space.

These related literatures show that the expression “Beatty shift” already had a broad semantic range before the symbolic-dynamical notion was formalized. The 2025 theory isolates a new meaning: a shift space whose geometry is controlled by two Beatty index maps and their overlap densities.

6. Conceptual significance, extensions, and open directions

The conceptual advance of the Beatty multiple shift is the replacement of rigid integer maps $m\ge 2$04 and $m\ge 2$05 by the more flexible maps

$m\ge 2$06

The result is a symbolic system whose dimension theory depends on fine distribution properties of Beatty sequences, not only on the combinatorics of a constraint matrix (Ban et al., 15 Jul 2025). The arithmetic of disjoint covering, overlap, and eventual synchronization becomes part of the dynamical invariant itself.

This places the theory adjacent to several other developments. Beatty-type formulas can solve nonlinear implicit recurrences: "Beatty solutions of almost Golomb equations" (Cloitre, 12 Apr 2026) proves that for every $m\ge 2$07 that is not an even perfect square, the order-$m\ge 2$08 Golomb equation admits a second monotone solution given by the inhomogeneous Beatty sequence

$m\ge 2$09

That result uses a window of $m\ge 2$10 shifted terms and shows that Beatty structure can arise in self-referential settings far removed from classical complementarity (Cloitre, 12 Apr 2026). This suggests a broader landscape in which Beatty constraints act as a unifying arithmetic mechanism across symbolic dynamics, recurrence theory, and probabilistic number theory.

Two misconceptions are worth isolating. First, a Beatty multiple shift is not merely a shifted Beatty sequence $m\ge 2$11; it is a shift space defined by a binary adjacency rule imposed along two Beatty-indexed coordinates. Second, the theory is not a small perturbation of affine multiple shifts. The integer-affine case is recovered exactly, but the general theory requires new arithmetic data, namely the density sequence $m\ge 2$12, and one parameter region remains unresolved (Ban et al., 15 Jul 2025).

The principal open problem presently recorded is to determine $m\ge 2$13 in region $m\ge 2$14 (Ban et al., 15 Jul 2025). A plausible implication is that further progress will require sharper Beatty-overlap classifications analogous to those used in disjoint covering theory. More broadly, the existing results indicate that the geometry of a Beatty multiple shift is encoded by the arithmetic of the two Beatty sequences that define it: once the overlap-density data are known, the Hausdorff and Minkowski dimensions follow by explicit formulas (Ban et al., 15 Jul 2025).