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An asymptotic for the growth of Markoff-Hurwitz tuples
Ryan Ronan (CUNY)
Location: Hill 525
Date & time: Tuesday, 27 February 2018 at 2:00PM - 3:00PM
The Markoff equation is a Diophantine equation in 3 variables first stud- ied in Markoff’s celebrated work on indefinite binary quadratic forms. An asymptotic formula to count the number of positive integral solutions to this equation below a given height was obtained by Zagier in 1982. In this talk, we establish an asymptotic formula for the growth of solutions to an n variable generalization of this equation, which we refer to as the Markoff- Hurwitz equation. The previous best result for n > 3 is due to Baragar in 1998 who established an exponential rate of growth with an exponent which is not, in general, an integer. We use methods from symbolic dynamics to improve this asymptotic count, and which yield a new interpretation of this exponent as the unique parameter for which there exists a certain conformal measure on projective space.
Joint work with Alex Gamburd and Michael Magee.