It is proved that the subword complexity $p_{T}(k)$ of $T$ is superlinear, $p_{T}(k)/k\to\infty$.
Abstract
Write $(3/2)^n = m_n + \varepsilon_n$ with $m_n$ the nearest integer and $\varepsilon_n\in[-\tfrac12,\tfrac12)$, and let $T=(t_n)$, $t_n=2m_{n+1}-3m_n$, be the resulting \emph{steering word}: the step-by-step record of the map $x\mapsto\tfrac32 x$ 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, $p_{T}(k)/k\to\infty$. 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.
Let $\Omega_n$ denote the set of $n\times n$ doubly stochastic matrices. Kim and Roush conjectured in 1981 that, for $n=2k+1>1$, $ \max_{A\in\Omega_{2k+1}}\operatorname{per}(I-A)=3\cdot 2^{k-2}$. They proposed the block construction $A_\star=\frac12(J_3-I_3)\oplus P_2^{\oplus(k-1)}$, where $P_2=\begin{pmatrix}0&1\\1&0\...
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Let $s(n)$ denote the number of ones in the binary expansion of an integer $n\in\mathbb{N}$, and let $\mu_t$ be the probability measure on $\mathbb{Z}$ defined by the asymptotic densities of the level sets of the function $\mathbb{N}\ni n\mapsto s(n+t)-s(n)\in\mathbb{Z}$. Let $P_t$ be the family of finitely supported m...
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For an integer $t\geq-1$, let $\theta_t$ be the largest real root of $g_t(X)=X^3-tX^2-(t+3)X-1$, and set $R_t=\mathbb{Z}[\theta_t]\subseteq\mathcal{O}_t=\mathcal{O}_{\mathbb{Q}(\theta_t)}$, $N_t=[\mathcal{O}_t:R_t]$, and $\varepsilon_t=[\mathcal{O}_t^\times:R_t^\times]$. We determine $\varepsilon_t$ when $N_t$ is squar...
For integers $k\le n$, let $L_{k,n}$ be the set of words over $[n]$ of length at most $k$ in which no symbol is repeated. We present a nondeterministic finite automaton (NFA) of size $3.918^k n^{O(1)}$, improving on the $4^{k+o(k)}n^{O(1)}$ construction of Ben-Basat, Gabizon, and Zehavi. Our proof organizes several cla...
Ran Ben Basat· 0 citations
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