Beyond Expansion IV: Traces of Thin Semigroups

- Mathematics, Institute for Advanced Study, Princeton
- More about Jean Bourgain

- Mathematics, Rutgers
- More about Alex Kontorovich

### Editorial introduction

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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