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This problem is closely connected with number-theoretic model for spin chains. In this model, to a finite chain of spins each of which can be directed upwards $(\uparrow)$ or downwards $(\downarrow)$, a product of the matrices $$ A=\begin{pmatrix} 1 & 0\\ 1 & 1 \end{pmatrix},\quad B=\begin{pmatrix} 1 & 1\\ 0 & 1 \end{pmatrix} $$ is assigned, according to the rule $\uparrow=A$, $\downarrow=B$. For example, $$ \uparrow\uparrow\uparrow\downarrow\downarrow\uparrow\uparrow\uparrow\uparrow= \uparrow^3\downarrow^2\uparrow^4=A^3B^2A^4. $$ The energy of a given configuration is $$ E(\uparrow^{a_1}\downarrow^{a_2}\uparrow^{a_3}\ldots)= \log\left(\mathrm{Tr}\,(A^{a_1}B^{a_2}A^{a_3}\ldots)\right). $$ Let $G$ be the free multiplicative monoid generated by the matrices $A$ and $B$. From a physical viewpoint, the asymptotic behaviour of the number of configurations $$ \Phi(N)=\bigl|\bigl\{C\in G: \mathrm{Tr}\,C=N \bigr\}\bigr|\qquad(N\ge 3) $$ with a given energy, and the number of configurations in which the energy does not exceed a given quantity, $$ \Psi(N)=\big |\big\{C\in G: 3\le\mathrm{Tr}\,C\leqslant N \big\}\big |= \sum_{3\le n\leqslant N}\Phi(n). $$ It is known, that $$ \Psi(N)=N^2(c_1\log N+c_0)+{O}(N^{3/2}\log^4N), $$ where $$ c_1=\frac{1}{\zeta(2)},\quad c_0= \frac{1}{\zeta(2)}\left(\gamma-\frac{3}{2}-\frac{\zeta'(2)} {\zeta(2)}\right). $$ And this result is a special case of a more general result concerning the Gauss-Kuz’min statistics for spin chains and Gauss-Kuz’min statistics for the quadratic irrationals. For more details see the paper Spin chains and Arnold's problem on the Gauss-Kuz'min statistics for quadratic irrationals..