Beyond Expansion IV: Traces of Thin Semigroups

Beyond Expansion IV: Traces of Thin Semigroups, Discrete Analysis 2018:6, 27 pp. This is the fourth in a series of papers by Bourgain and Kontorovich that study the arithmetic properties of groups and semigroups of 2-by-2 integer matrices. Given a finite set $A$ of natural numbers, $\Gamma_A$ is defined to be the semigroup generated by the $2\times 2$ matrices $M_a=\begin{pmatrix}a&1\\ 1&0\\ \end{pmatrix}$, where $a$ ranges over $A$. The semigroup $\Gamma_A$ is a subsemigroup of $\mathrm{GL}(2,\mathbb Z)$, and it is the focus of attention in this paper. If $\mathbf{a}=(a_1,...,a_n)$ is a tuple of natural numbers from $A$, let us denote by $M_{\mathbf a}$ the product $M_{a_1}...M_{a_n}$. Interest in the above semigroup is manifold, but it stems primarily from continued fractions, because it is easy to see that the quadratic irrational $[a_1,...,a_n]$ whose infinite periodic continued fraction expansion is given by $1 + a_1/(1+ a_2/( 1 + a_3/...))...$ is one of the two fixed points of the Möbius transformation of the real line induced by $M_{\mathbf a}$. There is also a beautiful geometric interpretation of the above in terms of geodesic loops on the modular surface $S$ (that is, the quotient of the hyperbolic plane by $\mathrm{PSL}(2,\mathbb Z)$). Indeed each geodesic loop corresponds to the conjugacy class of a matrix in $\mathrm{PSL}(2,\mathbb Z)$. When $A$ is the full set of all natural numbers, every matrix in $\mathrm{GL}(2,\mathbb Z)$ can be conjugated to a matrix in $\Gamma_A$ and the corresponding tuple $\mathbf a=(a_1,...,a_n)$ encodes how high in the cusp of $S$ this geodesic can venture. Also the quadratic irrational $[a_1,...,a_n]$ belongs to the splitting field of the characteristic polynomial of the matrix (which is a quadratic field). In previous papers, Bourgain and Kontorovich studied the family of integers that can arise as the top left entry of a matrix from the semigroup $\Gamma_A$. The celebrated Zaremba conjecture can be reformulated so as to state that every integer arises as the top left entry of some matrix from $\Gamma_A$ provided $A=\{1,...,N\}$ for some large enough integer $N$; probably N=5 is good enough. They showed that almost all integers can be represented this way. In this paper the authors study instead the set of integers that arise as _traces_ of elements from $\Gamma_A$. Each trace corresponds to a conjugacy class in $\mathrm{GL}(2,\mathbb R)$ and if all integers appear as such traces (for a given $A$), that implies that in every real quadratic field $\mathbb Q(\sqrt{D})$ there are infinitely many quadratic irrationals $[a_1,...,a_n]$ with all $a_i$s belonging to the fixed set $A$. Indeed if $t,s$ are integers satisfying the Pell-Fermat equation $t^2 - Ds^2 = 4$, then any matrix in $\Gamma_A$ with trace $t$ will produce such a quadratic irrational (as the fixed point of the associated Möbius transformation). In geometric terms this means that for each square-free $D$ there are infinitely many geodesics that stay in a fixed compact set of the modular surface and whose splitting field is $\mathbb Q(\sqrt{D})$. This is the Arithmetic Chaos conjecture of McMullen. Bourgain and Kontorovich thus conjecture that every large enough integer appears as the trace of an element from the semigroup $\Gamma_A$, provided that $A$ is large enough; it is possible that $A=\{1,2\}$ might suffice. Since there are exponentially many words of given length in $\Gamma_A$ and only linearly many traces, it is natural to expect even that each trace appears with very high, indeed exponential, multiplicity. In this paper they show a weaker result, which does not go as far as proving that such traces have full density, but which gives instead some interesting arithmetic information on the set of traces : they show that for almost any modulus $q$ the number of traces of products of at most $n$ matrices $M_a$, $a \in A$, that are divisible by $q$ is roughly equal to $1/q$ times the total number of traces of such products. And they show that the error term is small provided that $q$ is at most $n^{\alpha}$ for any $\alpha< 1/3$. They point out that previous techniques related to the Bourgain-Gamburd-Sarnak affine sieve easily yield a positive "level distribution alpha", but getting up to 1/3 requires new ideas that form the gist of this paper and are based on non-Abelian sieving techniques. Using results from [the PhD thesis of Paul Mercat](http://www.i2m.univ-amu.fr/perso/paul.mercat/Publis/these.pdf), they prove the following attractive application: if $A=\{1,...,50\}$ then the set of traces contains infinitely many numbers that have at most two prime factors. Finally there is a result of independent interest, which comes from the proof techniques: they have to study the additive energy of the set of matrices in $\mathrm{SL}(2,\mathbb Z)$ of norm at most $n$, and show that this additive energy is almost as small as it could possibly be. 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