PDE/Analysis Seminar

Speaker: Pavel Etingof(MIT) Title: q-continued fractions Abstract: Recall that every real number \(x \geq 1\) can be uniquely written as a continued fraction \[ x = \llbracket a_1, a_2, a_3, \dots \rrbracket := a_1 - \cfrac{1}{a_2 - \cfrac{1}{a_3 - \dots}} \] where the \(a_i\) are integers \(\geq 2\): namely, \(a_1\) is the smallest integer larger than \(x\), \(a_2\) is the smallest integer larger than \(\frac{1}{a_1-x}\), and so on. Moreover, the sequence of truncations of any such continued fraction converges to a real number in \([1,\infty)\). Recently, Morier-Genoud and Ovsienko defined a natural \(q\)-deformation of continued fractions, which associates to every nonnegative real number \(x\) a formal power series \([x]_q\) with integer coefficients, serving as a \(q\)-analog of \(x\): \[ [x]_q = \llbracket a_1, a_2, a_3, \dots \rrbracket_q := [a_1]_q - \cfrac{q^{a_1-1}}{[a_2]_q - \cfrac{q^{a_2-1}}{[a_3]_q - \dots}}, \] where \([n]_q = 1 + q + \dots + q^{n-1}\) for a positive integer \(n\). This extends the latter formula to all real \(x \geq 1\) and yields a rational function of \(q\) when \(x\) is rational. I will discuss the analytic properties of the \(q\)-real numbers \([x]_q\). Namely, I will explain why all these series converge for \(\lvert q \rvert < \frac{3-\sqrt{5}}{2}\), and why this bound is optimal, with equality for the golden ratio \(x = \frac{1+\sqrt{5}}{2}\) --- a \(q\)-analogue of Hurwitz's theorem stating that the golden ratio is the number worst-approximable by rationals. The proof uses a combinatorial description of the coefficients in terms of rooted trees. For real \(0 < q < 1\), \(q\)-continued fractions also define an increasing function \(x \mapsto [x]_q\), which exhibits jumps at rational points and has a derivative of zero almost everywhere. I will discuss its properties, as well as a complex analogue \([\tau]_q\) for \(\tau \in \mathbb{C}_+\), which is a meromorphic function in \(\tau\) expressed through hypergeometric and modular functions. The talk will include the necessary background and is based in part on joint work with Valentin Ovsienko. Based on arXiv:2508.08440 and arXiv:2608.06761.