Narrow Results

Based on the work of Boyd, we find all polynomials $A$ with integer coefficients and such that $A(0)\ge 1$ and $|A(z)|\le |1-2z|$ if $\left|z\right|=1$. Combining this fact with Amara’s results, we prove that there are only finitely many Pisot numbers $𝜃\in (1,2)$, with one of the Galois conjugates of $𝜃$ lying in the disc $\left|z\right|\le \frac{\sqrt{5}-1}{2}$. The truth of this sheds some new light on a recent conjecture of Hare and Sidorov.

This note considers a self-similar tiling of the complex plane generated by the minimal Pisot number β. It will show that the boundary of every tile in is a simple closed curve, hence each tile is homeomorphic to the unit disk.

Some kinds of the self-similar sets with overlapping structures are studied by introducing the graph-directed constructions satisfying the open set condition that coincide with these sets. In this way, the dimensions and the measures are obtained.

Let β be a fixed element of 𝔽_q((X^-1)) with polynomial part of degree ≥ 1, then any formal power series can be represented in base β, using the transformation T_β : f ↦ {βf} of the unit disk . Any formal power series in is expanded in this way into d_β(f) = (a_i(X))_i≥1, where . The main aim of this paper is to characterize the formal power series β(|β| > 1), such that d_β(1) is finite, eventually periodic or automatic (such characterizations do not exist in the real case).

Let $P(x):=a_d{x}^d+\cdots +a_0\in \mathbb{Q}[x]$, $a_d>0$, be a polynomial of degree $d\ge 2$. Let $(x_n)$ be a sequence of integers satisfying