Superlinear complexity of the (3/2)ⁿ steering word

2026/07/31 by Ralf Stephan
Mathematics · #math.NT #msc:11

paper · pdf

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arxiv created 2026/08/03 · arxiv updated 2026/08/04

Abstract

Write (3/2)ⁿ = m_n + ε_n with m_n the nearest integer and ε_n∈[-\tfrac12,\tfrac12), and let T=(t_n), t_n=2m_n+1-3m_n, be the resulting steering word: the step-by-step record of the map x↦\tfrac32 x on the orbit of 1, coded by nearest-integer rounding. Using results by Corvaja--Zannier and Nair--Kumar--Rout we prove that the subword complexity p_T(k) of T is superlinear, p_T(k)/k→∞. The argument is completely formalized in Lean~4 and rests on a single external input, the Evertse--Schlickewei S-arithmetic subspace theorem, from which both cited results are themselves derived within the formalization.

Citations

• Algebraic approximations to linear combinations of S-units
• On the Trace Subshifts of Fractional Multiplication Automata
• On the rational approximations to the powers of an algebraic number