Jun-ichi Tamura’s research while affiliated with Tsuda University and other places

We study four (families of) sets of algebraic integers of degree less than or equal to three. Apart from being simply defined, we show that they share two distinctive characteristics: almost uniformity and arithmetical independence. Here, ``almost uniformity'' means that the elements of a finite set are distributed almost equidistantly in the unit interval, while ``arithmetical independence'' means that the number fields generated by the elements of a set do not have a mutual inclusion relation each other. Furthermore, we reveal to what extent the algebraic number fields generated by the elements of the four sets can cover quadratic or cubic fields.

We give a new class of multidimensional p-adic continued fraction algorithms. We propose an algorithm in the class for which we can expect that multidimensional p-adic version of Lagrange's Theorem holds.

We present several continued fraction algorithms, each of which gives an eventually periodic expansion for every quadratic element of ${\mathbb Q}_p$ over ${\mathbb Q}$ and gives a finite expansion for every rational number. We also give, for each of our algorithms, the complete characterization of elements having purely periodic expansions.

We introduce a true orbit generation method enabling exact simulations of dynamical systems defined by arbitrary-dimensional piecewise linear fractional maps, including piecewise linear maps, with rational coefficients. This method can generate sufficiently long true orbits which reproduce typical behaviors (inherent behaviors) of these systems, by properly selecting algebraic numbers in accordance with the dimension of the target system, and involving only integer arithmetic. By applying our method to three dynamical systems—that is, the baker's transformation, the map associated with a modified Jacobi-Perron algorithm, and an open flow system—we demonstrate that it can reproduce their typical behaviors that have been very difficult to reproduce with conventional simulation methods. In particular, for the first two maps, we show that we can generate true orbits displaying the same statistical properties as typical orbits, by estimating the marginal densities of their invariant measures. For the open flow system, we show that an obtained true orbit correctly converges to the stable period-1 orbit, which is inherently possessed by the system.

We give a new algorithm of slow continued fraction expansion related to any real cubic number field as a 2-dimensional version of the Farey map. Using our algorithm, we can find the generators of dual substitutions (so-called tiling substitutions) for any stepped surface for any cubic direction.

There exist certain quadratic elements α∈ℚ((t −1)) over the rational function field ℚ(t) having nonperiodic continued fraction expansion, see W.M. Schmidt in (Acta Arith. 95(2):139–166, 2000). Hence we need a modification of Lagrange’s theorem with regard to function fields instead of number fields. In this paper, we introduce a class of continued fractions and describe Lagrange’s theorem as a conjecture related to quadratic elements over ℚ(t). We give some examples which support our conjecture.

In an earlier paper [Math. Comput. 78, No. 268, 2209–2222 (2009; Zbl 1217.11067)], we introduced a new algorithm which is something like the modified Jacobi-Perron algorithm, and gave some computer experiments by which we can expect that the expansion obtained by our algorithm for α ̲=(α 1 ,...,α s )∈K s (with some natural conditions on α) becomes periodic for any real number field K as far as s+1=deg ℚ (K)≤4. But, it seems very likely that the algorithm will not work well if deg ℚ (K)=5. In this paper we give a new algorithm and discuss some properties and experimental results by which we can expect that the expansion of αby the new algorithm always becomes periodic for any real number field K with deg ℚ (K)≤5.