标题： 非匹配和非周期性对于$(N,α)$展开 显示英文标题

标题： (non)-matching and (non)-periodicity for $(N,α)$-expansions

摘要： 最近，对于每个$N\in\mathbb{N}$、$N\geq 2$和$\alpha \in (0,\sqrt{N}-1]$，引入了一类新的连分数算法，即$(N,\alpha$-展开。 这些连分数算法中的每一个都只有有限多个可能的数字。 这些$(N,\alpha)$-展开与许多其他（经典）连分数算法表现出非常不同的行为。 在本文中，我们将证明，当数字集中的所有数字与$N$互质时，这发生在参数空间的指定区间内，会发生某种特殊的情况。 有理数和某些二次无理数将不会具有周期性展开。 此外，在这些区域中没有匹配区间。 这与常规连分数和更多经典的参数化连分数算法形成鲜明对比，这些算法通常被证明对几乎所有参数都成立。 另一方面，对于$\alpha$足够小，所有有理数都有一个最终周期为1的周期性展开。 当$N=2$时，这适用于所有$\alpha$。我们还发现无穷多个匹配区间对于$N=2$，以及不包含在任何匹配区间中的有理数。 显示英文摘要

摘要： Recently a new class of continued fraction algorithms, the $(N,\alpha$)-expansions, was introduced for each $N\in\mathbb{N}$, $N\geq 2$ and $\alpha \in (0,\sqrt{N}-1]$. Each of these continued fraction algorithms has only finitely many possible digits. These $(N,\alpha)$-expansions `behave' very different from many other (classical) continued fraction algorithms. In this paper we will show that when all digits in the digit set are co-prime with $N$, which occurs in specified intervals of the parameter space, something extraordinary happens. Rational numbers and certain quadratic irrationals will not have a periodic expansion. Furthermore, there are no matching intervals in these regions. This contrasts sharply with the regular continued fraction and more classical parameterised continued fraction algorithms, for which often matching is shown to hold for almost every parameter. On the other hand, for $\alpha$ small enough, all rationals have an eventually periodic expansion with period 1. This happens for all $\alpha$ when $N=2$. We also find infinitely many matching intervals for $N=2$, as well as rationals that are not contained in any matching interval.