arXiv subjects

Tadahisa Hamada

Publications and source records attributed to Tadahisa Hamada.

3 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

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↗

arXiv2024

A concise geometric proof of the three distance theorem

The three distance theorem states that for any given irrational number $α$ and a natural number $n$, when the interval $( 0, 1 )$ is divided into $n+1$ subintervals by integer multiples of $α$, namely, $\{0\}, \{ α\}, \{ 2\,α\},\dots, \{ n\,α\}$, then each subinterval is limited to at most three different lengths. Steinhaus conjectured this theorem in the 1950s, and many researchers have given various proofs since then. This paper aims to improve the perspective by showing a two-dimensional map which tells how the unit interval is divided by continuously changing $α$, and provide a concise proof of the theorem. By illustrating this proof through geometric visualizations, we offer a clearer and more intuitive understanding of the underlying principles and relationships. The approach not only reinforces the classical results but also paves the way for new insights and applications in the study of irrational numbers and their properties. Additionally, we present a simple proof of the three gap theorem, which is a dual of the three distance theorem.

math.NT↗