--- title: Beatty Multiple Shift Overview url: https://www.emergentmind.com/topics/beatty-multiple-shift type: topic --- # Beatty Multiple Shift Overview 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" [2507.10982], 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 [2507.10982]. ## 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 \(qk+b\) by the Beatty expressions \(\lfloor \alpha k+\beta\rfloor\) and \(\lfloor \gamma k+\delta\rfloor\), thereby extending the affine theory from integer parameters to real parameters [2507.10982]. This generalization is exact: if \((\alpha,\beta,\gamma,\delta)=(p,a,q,b)\in\mathbb Z^4\), then \[ X_A^{[\alpha,\beta,\gamma,\delta]}=X_A^{\langle p,a,q,b\rangle}. \] 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 \(p=1\) and \(a=b=0\) [2507.10982]. 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 \[ \lfloor \alpha k+\beta\rfloor \quad\text{and}\quad \lfloor \gamma k+\delta\rfloor. \] ## 2. Arithmetic decomposition by Beatty overlap The arithmetic core of the theory is the pair of sets \[ S(\alpha,\beta):=\{\lfloor \alpha k+\beta\rfloor:k\in\mathbb N\}\cap\mathbb N, \qquad S(\gamma,\delta):=\{\lfloor \gamma k+\delta\rfloor:k\in\mathbb N\}\cap\mathbb N. \] The paper defines a map \[ f\!\left(\lfloor \alpha k+\beta\rfloor\right)=\lfloor \gamma k+\delta\rfloor, \] which is well-defined because different \(k\) cannot produce the same \(\lfloor \gamma k+\delta\rfloor\) when \(\gamma\ge 1\) [2507.10982]. The positive integers are then decomposed into classes \(A_i\), indexed by \(i\in\mathbb N\cup\{\infty\}\), according to how the orbit under \(f\) interacts with the overlap of the two Beatty sets. The description given in [2507.10982] is as follows. \(A_1\) consists of integers in neither sequence. \(A_2\) consists of points that move from \(S(\alpha,\beta)\setminus S(\gamma,\delta)\) to \(S(\gamma,\delta)\setminus S(\alpha,\beta)\) in one step. For \(i\ge 3\), \(A_i\) contains points that remain in the intersection for \(i-2\) steps and then exit. The set \(A_\infty\) contains points that remain forever in the intersection under iteration. For each \(i\in\mathbb N\cup\{\infty\}\), let \(d_i\) be the asymptotic density of \(A_i\). The sequence \[ \mathbf d=(d_1,d_2,d_3,\dots,d_\infty) \] 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 [2507.10982]. 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 [2507.10982] gives explicit dimension formulas in terms of the density sequence \(\mathbf d\). Assume that \(A\) is an \(m\times m\) irreducible binary matrix, that \(1\le \alpha<\gamma\), that \(\beta,\delta\in\mathbb R\), and that all densities \(d_i\) exist. Under these hypotheses, the Minkowski dimension of \(X_A^{[\alpha,\beta,\gamma,\delta]}\) is given by an explicit series involving \(\mathbf d\), the ratio \(\gamma/\alpha\), and the quantities \(|A^{i-1}|\). If \(A\) is primitive, the Hausdorff dimension is also given explicitly, through a formula involving \(\mathbf d\) and a unique positive vector \((t_j)_{j=0}^{m-1}\) satisfying \[ t_i^{\gamma/\alpha}=\sum_{j=0}^{m-1} A(i,j)t_j. \] As stated in the paper, this is the generalized eigenvector relation appearing in the thermodynamic formalism for these shifts [2507.10982]. The interpretation of the terms is explicit. The coefficients \(d_i\) measure how much of the integer set falls into each Beatty dynamical regime. The factor \((\gamma/\alpha)^{i-1}\) reflects the geometric growth of the index map \(f\), since \(f\) expands roughly by \(\gamma/\alpha\). The quantities \(|A^{i-1}|\) count admissible words of length \(i\) along a length-\(i-1\) constraint chain. In the Hausdorff formula, \(d_1\) corresponds to indices with no constraint propagation, the coefficients \(d_i\) for \(i\ge 2\) correspond to finite propagation depth, and \(d_\infty\) measures the contribution of infinite propagation [2507.10982]. A further structural criterion is inherited from earlier multiple-shift theories: Hausdorff and Minkowski dimensions coincide if and only if the matrix \(A\) has equal row sums [2507.10982]. 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 [2507.10982] divides the parameter space into regions \(\langle 1\rangle,\dots,\langle 6\rangle\), and in several of these regions the density sequence \(\mathbf d\) is computed explicitly. | Region | Density sequence \(\mathbf d\) | Status | |---|---|---| | \(\langle 1\rangle\) | \((0,0,\underline 0,\,1-\tfrac1\gamma)\) | explicit | | \(\langle 3\rangle\cup\langle 4\rangle\) | \(\left(\frac{(\alpha-1)(\gamma-1)}{\alpha\gamma},\frac{(\alpha-1)(\gamma-1)}{\alpha^2\gamma},\left(\frac{(\alpha-1)(\gamma-1)}{\alpha^i\gamma}\right)_{i\ge 3},0\right)\) | explicit | | \(\langle 5\rangle\) | \(\left(1-\frac1\alpha-\frac1\gamma,\frac1\alpha,\underline 0,0\right)\) | explicit | | \(\langle 2\rangle\) | separate arithmetic description in the appendix | partial explicitness | | \(\langle 6\rangle\) | determine \(\mathbf d=(d_1,d_2,d_3,\dots,d_\infty)\) | open | Region \(\langle 6\rangle\) is singled out by an explicit open problem: determine the full sequence \(\mathbf d=(d_1,d_2,d_3,\dots,d_\infty)\) for parameters in that region [2507.10982]. 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 \(\langle 7\rangle,\langle 8\rangle,\langle 9\rangle,\langle 10\rangle\), 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 [2507.10982]. 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 \(\alpha,\gamma,\beta,\delta\) 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 \[ S(\alpha,\beta):=\bigl(\lfloor n\alpha+\beta\rfloor\bigr)_{n\in\mathbb Z}, \] and the shift parameter \(\beta\) has long been used to study complementarity and partition phenomena. In the stadium model of "A Model for Pairs of Beatty Sequences" [1103.5198], the shifts encode starting positions of two athletes. For irrational \(\alpha_1,\alpha_2\) with \[ \frac1{\alpha_1}+\frac1{\alpha_2}=1, \] Skolem’s theorem is recalled in the form that \(S(\alpha_1,\beta_1)\) and \(S(\alpha_2,\beta_2)\) eventually partition \(\mathbb Z\) if and only if \[ \beta_1+\beta_2\in\mathbb Z. \] That literature uses “shift” in the non-homogeneous Beatty sense rather than in the symbolic-dynamical sense [1103.5198]. A second strand concerns multiple shifted Beatty sequences in analytic number theory. In "Multivariate and quantitative Erdős-Kac laws for Beatty sequences" [2602.04875], the objects are \[ \omega(\lfloor \alpha_i n+\beta_i\rfloor), \] and the main theorem states that for \(k\ge 1\), if \(\alpha_i/\alpha_j\notin\mathbb Q\) for all \(i\neq j\), then the centered and normalized vector converges in distribution to the standard \(k\)-dimensional Gaussian. The same paper proves that for any real \(\alpha>0\) and \(\beta\in\mathbb R\), the Kolmogorov distance for a single Beatty sequence is bounded by \[ O\!\left(\frac{\log\log\log N}{\sqrt{\log\log N}}\right), \] while no parameter-independent quantitative rate exists in general for joint convergence across multiple Beatty sequences [2602.04875]. A third strand studies intersections of multiple Beatty sequences with prime-rich sets. "Piatetski-Shapiro primes in the intersection of multiple Beatty sequences" [2109.00461] proves that if \(\alpha_1,\dots,\alpha_\xi>1\) are irrational numbers of finite type and \(1,\alpha_1^{-1},\dots,\alpha_\xi^{-1}\) are linearly independent over \(\mathbb Q\), then for \(1