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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 $$ E(\uparrow^{a_1}\downarrow^{a_2}\uparrow^{a_3}\dotsm)= \log\left(\operatorname{Tr}(A^{a_1}B^{a_2}A^{a_3}\dotsm)\right). $$ Let $G$ be the free multiplicative monoid generated by the matrices $A$ and $B$. From a physical point of view it is interesting to know the asymptotic behaviour of the number of configurations $$ \Phi(N)=\bigl|\bigl\{C\in G: \operatorname{Tr} C=N \bigr\}\bigr|\qquad(N\ge 3) $$ with a given energy, and the number of configurations in which the energy does not exceed a given quantity, $$ \Psi(N)=\bigl |\bigl\{C\in G: 3\le\operatorname{Tr} C\leqslant N \bigr\}\bigr |= \sum_{3\le n\leqslant N}\Phi(n). $$ 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 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.