Alphabets, rewriting trails and periodic representations in algebraic bases

For $$\beta > 1$$ a real algebraic integer (the base), the finite alphabets $${\mathcal {A}} \subset {\mathbb {Z}}$$ which realize identity $${\mathbb {Q}}(\beta ) = \mathrm{Per}_{{\mathcal {A}}}(\beta )$$ , where $$\mathrm{Per}_{{\mathcal is set of complex numbers are $$(\beta {\mathcal {A}})$$ -eventually periodic representations, investigated. Comparing with greedy algorithm, minimal and natural defined. The shown to be correlated asymptotics Pierce base $$ Lehmer’s problem. notion rewriting trail introduced construct intermediate associated small polynomial values base. Consequences on representations neighbourhoods origin in generalizing Schmidt’s theorem related Pisot numbers, Applications Galois conjugation given for convergent sequences bases $$\gamma _s := \gamma _{n, m_1 \ldots m_s}$$ such that _{s}^{-1}$$ unique root (0, 1) an almost Newman type $$-1+x+x^n +x^{m_1}+\cdots + x^{m_s}$$ $$n \ge 3$$ $$s $$m_1 - n n-1$$ $$m_{q+1}-m_q all $$q . reciprocal close one, poles modulus $$< dynamical zeta function -shift $$\zeta _{\beta }(z)$$ shown, under some assumptions, zeroes