arXiv subjects
Publications and source records attributed to Shigeki Akiyama.
We give asymptotic formulas for the number of balanced words whose slope $α$ and intercept $ρ$ lie in a prescribed rectangle. They are related to uniform distribution of Farey fractions and Riemann Hypothesis. In the general case, the error term is deduced using an inequality of large sieve type.
We study a polyhedron with $n$ vertices of fixed volume having minimum surface area. Completing the proof of Fejes Toth, we show that all faces of a minimum polyhedron are triangles, and further prove that a minimum polyhedron does not allow deformation of a single vertex. We also present possible minimum shapes for $n\le 12$, some of them are quite unexpected, in particular $n=8$.
We consider primitive substitution tilings on R^d whose expansion maps are unimodular. We assume that all the eigenvalues of the expansion maps are algebraic conjugates with the same multiplicity. In this case, we can construct a cut-and-project scheme with a Euclidean internal space. Under some additional condition, we show that if the substitution tiling has pure discrete spectrum, then the corresponding representative point sets are regular model sets in that cut-and-project scheme.
Let $β>1$ be an integer or generally a Pisot number. Put $T(x) = \{ βx \}$ on $[0,1]$ and let $S: [0,1]\to [0,1]$ be a piecewise linear transformation whose slopes have the form $\pm β^m$ with positive integers $m$. We give sufficient conditions that $T$ and $S$ have the same generic points.
We show that a constant angle progression on the Fermat spiral forms a Delone set if and only if its angle is badly approximable.
We give a sufficient geometric condition for a subshift to be measurably isomorphic to a domain exchange and to a translation on a torus. And for an irreducible unit Pisot substitution, we introduce a new topology on the discrete line and we give a simple necessary and sufficient condition for the symbolic system to have pure discrete spectrum. This condition gives rise to an algorithm based on computation of automata. To see the power of this criterion, we provide families of substitutions that satisfies the Pisot substitution conjecture: 1) $a \mapsto a^kbc$, $b \mapsto c$, $c \mapsto a$, for $k \in \mathbb{N}$ and 2) $a \mapsto a^lba^{k-l}$, $b \mapsto c$, $c \mapsto a$, for $k \in \mathbb{N}_{\geq 1}$, for $0 \leq l \leq k$ using different methods. And we also provide an example of $\mathcal{S}$-adic system with pure discrete spectrum everywhere.
We study the limit shape of successive coronas of a tiling, which models the growth of crystals. We define basic terminologies and discuss the existence and uniqueness of corona limits, and then prove that corona limits are completely characterized by directional speeds. As an application, we give another proof that the corona limit of a periodic tiling is a centrally symmetric convex polyhedron (see [Zhuravlev 2001], [Maleev-Shutov 2011]).
Every rational number p/q defines a rational base numeration system in which every integer has a unique finite representation, up to leading zeroes. This work is a contribution to the study of the set of the representations of integers. This prefix-closed subset of the free monoid is naturally represented as a highly non-regular tree. Its nodes are the integers, its edges bear labels taken in {0,1,...,p-1}, and its subtrees are all distinct. We associate with each subtree (or with its root n) three infinite words. The bottom word of n is the lexicographically smallest word that is the label of a branch of the subtree. The top word of n is defined similarly. The span-word of n is the digitwise difference between the latter and the former. First, we show that the set of all the span-words is accepted by an infinite automaton whose underlying graph is essentially the same as the tree itself. Second, we study the function that computes for all n the bottom word associated with n+1 from the one associated with n, and show that it is realised by an infinite sequential transducer whose underlying graph is once again essentially the same as the tree itself. An infinite word may be interpreted as an expansion in base p/q after the radix point, hence evaluated to a real number. If T is a subtree whose root is n, then the evaluations of the labels of the branches of T form an interval of $\mathbb{R}$. The length of this interval is called the span of n and is equal to the evaluation of the span-word of n. The set of all spans is then a subset of R and we use the preceding construction to study its topological closure. We show that it is an interval when p is greater than or equal to 2q-1, and a Cantor set of measure zero otherwise.
We give an expression for the Garsia entropy of Bernoulli convolutions in terms of products of matrices. This gives an explicit rate of convergence of the Garsia entropy and shows that one can calculate the Hausdorff dimension of the Bernoulli convolution $ν_β$ to arbitrary given accuracy whenever $β$ is algebraic. In particular, if the Garsia entropy $H(β)$ is not equal to $\log(β)$ then we have a finite time algorithm to determine whether or not $\mathrm{dim}_\mathrm{H} (ν_β)=1$.
We consider the self-affine tiles with collinear digit set defined as follows. Let $A,B\in\mathbb{Z}$ satisfy $|A|\leq B\geq 2$ and $M\in\mathbb{Z}^{2\times2}$ be an integral matrix with characteristic polynomial $x^2+Ax+B$. Moreover, let $\mathcal{D}=\{0,v,2v,\ldots,(B-1)v\}$ for some $v\in\mathbb{Z}^2$ such that $v,M v$ are linearly independent. We are interested in the topological properties of the self-affine tile $\mathcal{T}$ defined by $M\mathcal{T}=\bigcup_{d\in\mathcal{D}}(\mathcal{T}+d)$. Lau and Leung proved that $\mathcal{T}$ is homeomorphic to a closed disk if and only if $2|A|\leq B+2$. In particular, $\mathcal{T}$ has no cut point. We prove here that $\mathcal{T}$ has a cut point if and only if $2|A|\geq B+5$. For $2|A|-B\in \{3,4\}$, the interior of $\mathcal{T}$ is disconnected and the closure of each connected component of the interior of $\mathcal{T}$ is homeomorphic to a closed disk.
We complete statement and proof for B. Mossé's unilateral recognizability theorem. We also provide an algorithm for deciding the unilateral non-recognizability of a given primitive substitution.
It is known that the Möbius function in number theory is higher order oscillating. In this paper we show that there is another kind of higher order oscillating sequences in the form $(e^{2πi αβ^{n}g(β)})_{n\in \N}$, for a non-decreasing twice differentiable function $g$ with a mild condition. This follows the result we prove in this paper that for a fixed non-zero real number $α$ and almost all real numbers $β>1$ (alternatively, for a fixed real number $β>1$ and almost all real numbers $α$) and for all real polynomials $Q(x)$, sequences $\big(αβ^{n}g(β)+Q(n)\big)_{n\in \N}$ are uniformly distributed modulo $1$.
We study invariant measures of a piecewise expanding map in $\mathbb{R}^m$ defined by an expanding similitude modulo lattice. Using the result of Bang on a problem of Tarski, we show that when the similarity ratio is not less than $m+1$, it has an absolutely continuous invariant measure equivalent to the $m$-dimensional Lebesgue measure, under some mild assumption on the fundamental domain. Applying the method to the case $m=2$, we obtain an alternative proof of the result in Akiyama-Caalim:2015 together with some improvement.
A nearly linear recurrence sequence (nlrs) is a complex sequence $(a_n)$ with the property that there exist complex numbers $A_0$,$\ldots$, $A_{d-1}$ such that the sequence $\big(a_{n+d}+A_{d-1}a_{n+d-1}+\cdots +A_0a_n\big)_{n=0}^{\infty}$ is bounded. We give an asymptotic Binet-type formula for such sequences. We compare $(a_n)$ with a natural linear recurrence sequence (lrs) $(\tilde{a}_n)$ associated with it and prove under certain assumptions that the difference sequence $(a_n- \tilde{a}_n)$ tends to infinity. We show that several finiteness results for lrs, in particular the Skolem-Mahler-Lech theorem and results on common terms of two lrs, are not valid anymore for nlrs with integer terms. Our main tool in these investigations is an observation that lrs with transcendental terms may have large fluctuations, quite different from lrs with algebraic terms. On the other hand we show under certain hypotheses, that though there may be infinitely many of them, the common terms of two nlrs are very sparse. The proof of this result combines our Binet-type formula with a Baker type estimate for logarithmic forms.
We study a family of piecewise expanding maps on the plane, generated by composition of a rotation and an expansive similitude of expansion constant $β$. We give two constants $B_1$ and $B_2$ depending only on the fundamental domain that if $β>B_1$ then the expanding map has a unique absolutely continuous invariant probability measure, and if $β>B_2$ then it is equivalent to $2$-dimensional Lebesgue measure. Restricting to a rotation generated by $q$-th root of unity $ζ$ with all parameters in $\mathbb{Q}(ζ,β)$, it gives a sofic system when $\cos(2π/q) \in \mathbb{Q}(β)$ and $β$ is a Pisot number. It is also shown that the condition $\cos(2π/q) \in \mathbb{Q}(β)$ is necessary by giving a family of non-sofic systems for $q=5$.
We show that strong coincidences of a certain many choices of control points are equivalent to overlap coincidence for the suspension tiling of Pisot substitution. The result is valid for degree $\ge 2$ as well, under certain topological conditions. This result gives a converse of the paper by Akiyama-Lee and elucidates the tight relationship between two coincidences.
By the algorithm implemented in the paper [2] by Akiyama-Lee and some of its predecessors, we have examined the pure discreteness of the spectrum for all irreducible Pisot substitutions of trace less than or equal to $2$, and some cases of planar tilings generated by boundary substitutions due to the paper [17] by Kenyon.
A complex number $α$ is said to satisfy the height reducing property if there is a finite set $F\subset \mathbb{Z}$ such that $\mathbb{Z}[α]=F[α]$, where $\mathbb{Z}$ is the ring of the rational integers. It is easy to see that $α$ is an algebraic number when it satisfies the height reducing property. We prove the relation $\operatorname{Card}(F)\geq \max\{2,\left\vert M_α(0)\right\vert \},$ where $M_α$ is the minimal polynomial of $α$ over the field of the rational numbers, and discuss the related optimal cases, for some classes of algebraic numbers $α$. In addition, we show that there is an algorithm to determine the minimal height polynomial of a given algebraic number, provided it has no conjugate of modulus one.