Terminal Blocks of Primes in Pisot Numeration Systems

Abstract

We prove a prime number theorem for fixed terminal block words in Pisot integer numeration systems. If the dominant root $φ$ is a Pisot number and the characteristic polynomial $P_h$ is its minimal polynomial, every terminal block word of total digit-length $m$ occurs among the primes with asymptotic frequency $φ^{-m}$. In the Zeckendorf case this resolves a recent conjecture. The proof converts terminal conditions into Rauzy cylinder windows and then into a linear orbit on a compact torus. For these companion substitutions, the required multiplicity-one Rauzy geometry is automatic: Barge's pure-discreteness theorem applies after reversal of the substitution words. The Rauzy torus also gives a prime number theorem for the substitution fixed word: every finite factor occurs at prime starting positions with its ordinary factor frequency. This proves the Tribonacci prime-number theorem suggested by Drmota--Müllner--Spiegelhofer. The toral model further yields polynomial sampling laws, asymptotic independence from fixed congruence classes, fixed-shift correlation formulas, and Möbius orthogonality. Combined with established prime theorems, it gives a terminal refinement of Chebotarev and shows that every fixed terminal prime class contains arbitrarily long arithmetic progressions with polylogarithmically bounded common difference.