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Shigeki Akiyama

Publications and source records attributed to Shigeki Akiyama.

At least 19 recordsLinked to original sources
arXiv2026

Sturmian lattices and Aperiodic tile sets

We give an explicit algorithm to construct aperiodic tile sets based on Sturmian words of quadratic slopes. The method works for any quadratic irrational slope, and we can produce infinitely many aperiodic tile sets whose underlying scaling constant is a unit of any real quadratic field. There are two key ingredients in our construction. The first one is ``Sturmian lattices''; an interesting grid structure generated by Sturmian words that emerged in an aperiodic monotile called Smith Turtle. We shall give a classification of Sturmian lattices. The second is the bounded displacement equivalence of Delone sets, which plays a central role in this construction.

math.CO↗
arXiv2026

The natural extension of the $(-β)$-transformation

We give a concrete construction of a natural extension of $(-β)$-transformation when $β$ is greater than the golden mean. Our construction relies on its Markov diagram and the eigenvectors of the associated countable Markov shifts. Its positive recurrence can be shown by path counting using a special property of the diagram. Our down-to-earth construction elucidates the result of Bruin-Kalle \cite{BruinKalle} by examples.

math.DS↗
arXiv2026

Aperiodic tile sets from Sturmian lattices

We give an explicit algorithm to construct aperiodic tile sets based on Sturmian words of quadratic slopes. The method works for any quadratic irrational slope, and we can produce an aperiodic tile set whose underlying scaling constant is a unit of any real quadratic field. There are two key ingredients in our construction. The first one is the ``Sturmian lattices'', an interesting grid structure generated by Sturmian words that emerged in an aperiodic monotile called Smith Turtle. The second is the bounded displacement equivalence of Delone sets, which plays a central role in this construction. A classification of Sturmian lattices and complete proofs are given in the full version.

math.CO↗
arXiv2026

Dynamical systems defined by polynomials with algebraic properties

Let (x_n; n\in Z) be a bisequence of elements x_n in the 1-dimensional torus R/Z, which is called a stream over R/Z. Let P(z)=a_k z^k+...+a_1 z+a_0 be a polynomial with integer coefficients. Define the set of streams over R/Z such that the convolution product P(z)\times(x_n; n\in Z)=(\sum_{i=0}^k a_i x_{n-i}; n\in Z)=(0; n\in Z), which is called the stream 0 of P. We study similarities between stream 0 of P and the roots of P(z)=0.

math.NT↗
arXiv2025

Overlapping substitutions and tilings

We generalize the notion of (geometric) substitution rule to obtain overlapping substitutions. Our motivating example is the substitution presented in Ziherl, Dotera and Bekku \cite{DBZ}, which features a substitution matrix with non-integer entries. We give the meaning of such a matrix by showing that the right Perron--Frobenius eigenvector encodes the patch frequency of the resulting tiling. The patch frequencies are shown to be uniformly convergent, implying that the corresponding dynamical system is uniquely ergodic. Under mild assumptions, we further prove that the associated expansion constant is always an algebraic integer. In general, overlapping substitutions may yield a patch with illegal (partial) overlaps of tiles, even if it is locally consistent. We provide a sufficient condition for an overlapping substitution to be consistent, ensuring that no such illegal tiles emerge. Finally, we construct many intriguing one-dimensional overlapping substitutions and present higher dimensional examples from Delone multi-sets with inflation symmetry.

math.CO↗
arXiv2025

Non-self-intersective dragon curves

Let us fold a strip of paper many times in the same direction, and then unfold it to form a fixed angle $θ$ at all creases. The resulting shape is called the Dragon curve with the unfolding angle $θ$. When $0\leθ<90^{\circ}$, the corresponding Dragon curve has a self-intersection. When $θ=180^{\circ}$, the corresponding Dragon curve is a straight line, which has no self-intersection. In this paper, we will show that any Dragon curve whose unfolding angle is greater than $99.3438^{\circ}$ and less than $180^{\circ}$ has no self-intersection.

math.MG↗
arXiv2025

Delone sets associated with badly approximable triangles

We construct new Delone sets associated with badly approximable numbers which are expected to have rotationally invariant diffraction. We optimize the discrepancy of corresponding tile orientations by investigating the linear equation $x+y+z=1$ where $πx$, $πy$, $πz$ are three angles of a triangle used in the construction and $x$, $y$, $z$ are badly approximable. In particular, we show that there are exactly two solutions that have the smallest partial quotients by lexicographical ordering.

math.NT↗
arXiv2025

Exponential Diophantine approximation and symbolic dynamics

We extend the key formula which intertwines multiplicative Markoff-Lagrange spectrum and symbolic dynamics. The proof uses complex analysis and elucidates the strategy of the problem. Moreover, the new method applies to a wide variety of polynomials possibly having multiple roots. We derive several consequences of this formula, which are expected on the Markoff-Lagrange spectrum.

math.NT↗
arXiv2024

3 dimensional Wythoff Nim

We introduce a new generalization of Wythoff Nim using three piles of stones. We show that its P-positions have finite difference properties and produce a partition of positive integers. Further, we give a conjecture that the P-positions approximate a half-line whose slope is described by algebraic numbers of degree 5.

math.CO↗
arXiv2024

Self-descriptive Sequences directed by two Periodic Sequences

In the present work, we exhibit a class of self-descriptive sequences that can be explicitly computed and whose frequencies are known. In particular, as a corollary of our main result, we prove that the sequence introduced in \citeBJM23 has the expected frequencies of occurrences.

cs.FL↗
arXiv2024

Intersection the twin dragon with rational lines

The Knuth Twin Dragon is a compact subset of the plane with fractal boundary of Hausdorff dimension $s = (\log λ)/(\log \sqrt{2})$, $λ^3 = λ^2 + 2$. Although the intersection with a generic line has Hausdorff dimension $s-1$, we prove that this does not occur for lines with rational parameters. We further describe the intersection of the Twin Dragon with the two diagonals as well as with various axis parallel lines.

math.MG↗
arXiv2022

Curious congruences for cyclotomic polynomials

Let $Φ_n^{(k)}(x)$ be the $k$-th derivative of $n$-th cyclotomic polynomial. Extending a work of D.~H.~Lehmer, we show some curious congruences: $2Φ^{(3)}_n(1)$ is divisible by $ϕ(n)-2$ and $Φ^{(2k+1)}_n(1)$ is divisible by $ϕ(n)-2k$ for $k\ge 2$. The congruence stems from a general property of self-reciprocal polynomials.

math.NT↗
arXiv2022

Periodic expansion of one by Salem numbers

We show that for a Salem number $β$ of degree $d$, there exists a positive constant $c(d)$ that $β^m$ is a Parry number for integers $m$ of natural density $\ge c(d)$. Further, we show $c(6)>1/2$ and discuss a relation to the discretized rotation in dimension $4$.

math.NT↗
arXiv2022

Width deviation of convex polygons

We consider the width $X_T(ω)$ of a convex $n$-gon $T$ in the plane along the random direction $ω\in\mathbb{R}/2π\mathbb{Z}$ and study its deviation rate: $$ δ(X_T)=\frac{\sqrt{\mathbb{E}(X^2_T)-\mathbb{E}(X_T)^2}}{\mathbb{E}(X_T)}. $$ We prove that the maximum is attained if and only if $T$ degenerates to a $2$-gon. Let $n\geq 2$ be an integer which is not a power of $2$. We show that $$ \sqrt{\fracπ{4n\tan(\fracπ{2n})} +\frac{π^2}{8n^2\sin^2(\fracπ{2n})}-1} $$ is the minimum of $δ(X_T)$ among all $n$-gons and determine completely the shapes of $T$'s which attain this minimum. They are characterized as polygonal approximations of equi-Reuleaux bodies, found and studied by K.~Reinhardt. In particular, if $n$ is odd, then the regular $n$-gon is one of the minimum shapes. When $n$ is even, we see that regular $n$-gon is far from optimal.We also observe an unexpected property of the deviation rate on the truncation of the regular triangle.

math.PR↗
arXiv2021

Multiplicative analogue of Markoff-Lagrange spectrum and Pisot numbers

Markoff-Lagrange spectrum uncovers exotic topological properties of Diophantine approximation. We investigate asymptotic properties of geometric progressions modulo one and observe significantly analogous results on the set \[ {\mathcal L}(α)=\left\{\left.\limsup_{n\to \infty}\|ξα^n\|\ \right|\ ξ\in {\mathbb R}\right\}, \] where $\|x\|$ is the distance from $x$ to the nearest integer. First, we show that ${\mathcal L}(α)$ is closed in $[0,1/2]$ for any Pisot number $α$. Then we consider the case where $α$ is an integer with $α\geq 2$, or a quadratic unit with $α\ge 3$. We show that ${\mathcal L}(α)$ contains a proper interval when $α$ is quadratic but it does not when $α$ is an integer. We also determine the minimum limit point and all isolated points beneath this point. In the course of the proof, we revisit a property studied by Markoff which characterizes bi-infinite balanced words and sturmian words.

math.NT↗
arXiv2021

On arithmetic progressions in non-periodic self-affine tilings

We study the repetition of patches in self-affine tilings in R^d. In particular, we study the existence and non-existence of arithmetic progressions. We first show that an arithmetic condition of the expansion map for a self-affine tiling implies the non-existence of certain one-dimensional arithmetic progressions. Next, we show that the existence of full-rank infinite arithmetic progressions, pure discrete dynamical spectrum, and limit periodicity are all equivalent for a certain class of self-affine tilings. We finish by giving a complete picture for the existence/non-existence of full-rank infinite arithmetic progressions in the self-similar tilings in R^d.

math.DS↗