Terminal Blocks of Primes in Pisot Numeration Systems
Abstract
We prove a prime number theorem for fixed terminal block words in a family of Pisot integer numeration systems defined by canonical block expansions. Suppose that the dominant root φ is a Pisot number and that the characteristic polynomial Ph is its minimal polynomial. Then every terminal block word of total digit-length m occurs among the primes with asymptotic frequency φ−m. The result recovers the known Zeckendorf and generalized Fibonacci cases and extends the terminal-prime problem to irreducible Pisot-unit systems without assuming that the recurrence coefficients are monotone, that the canonical expansions are greedy, or that the finiteness property (F) holds. Explicit non-β families and Pisot units failing (F) illustrate this extension. The main structural ingredient is a canonical block language whose expansion of the companion substitution fixed word records the corresponding integer expansions in natural order. This connects terminal conditions with Rauzy cylinder windows and then with a linear orbit on a compact torus. The required multiplicity-one geometry follows from Barge's pure-discreteness theorem after reversal of the substitution words. The same toral model yields prime genericity for finite factors of the substitution fixed word, polynomial sampling laws, asymptotic independence from fixed congruence classes, fixed-shift correlation formulas, and Möbius orthogonality. Combined with established prime theorems, it also gives a terminal refinement of Chebotarev and arithmetic progressions inside fixed terminal prime classes.
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BibTeX
@misc{chang2026terminal,
title = {{Terminal Blocks of Primes in Pisot Numeration Systems}},
author = {Sungkon Chang and Johann Verwee},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10026},
eprint = {2610.10026},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10026},
}
RIS
TY - UNPB
TI - Terminal Blocks of Primes in Pisot Numeration Systems
AU - Sungkon Chang
AU - Johann Verwee
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.10026
UR - https://arxiv.org/abs/2610.10026
AB - We prove a prime number theorem for fixed terminal block words in a family of Pisot integer numeration systems defined by canonical block expansions. Suppose that the dominant root φ is a Pisot number and that the characteristic polynomial P_h is its minimal polynomial. Then every terminal block word of total digit-length m occurs among the primes with asymptotic frequency φ^−m. The result recovers the known Zeckendorf and generalized Fibonacci cases and extends the terminal-prime problem to irreducible Pisot-unit systems without assuming that the recurrence coefficients are monotone, that the canonical expansions are greedy, or that the finiteness property (F) holds. Explicit non-β families and Pisot units failing (F) illustrate this extension. The main structural ingredient is a canonical block language whose expansion of the companion substitution fixed word records the corresponding integer expansions in natural order. This connects terminal conditions with Rauzy cylinder windows and then with a linear orbit on a compact torus. The required multiplicity-one geometry follows from Barge's pure-discreteness theorem after reversal of the substitution words. The same toral model yields prime genericity for finite factors of the substitution fixed word, polynomial sampling laws, asymptotic independence from fixed congruence classes, fixed-shift correlation formulas, and Möbius orthogonality. Combined with established prime theorems, it also gives a terminal refinement of Chebotarev and arithmetic progressions inside fixed terminal prime classes.
ER -