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, 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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