I will review how the equations of general relativity near a spacetime singularity map onto an arithmetic hyperbolic billiard dynamics. The semiclassical quantum states for this dynamics are Maaβ cusp forms on fundamental domains of modular groups. For example, gravity in four spacetime dimensions leads to PSL(2,Z) while five dimensional gravity leads to PSL(2,Z[w]), with Z[w] the Eisenstein integers. The automorphic forms can be expressed, in a dilatation (Mellin transformed) basis as L-functions. The Euler product representation of these L-functions indicates that these quantum states admit a dual interpretation as a "primon gas" partition function. I will describe some physically motivated mathematical questions that arise from these observations.

An old conjecture of Graham asks whether there are infinitely many integers n such that \binom{2n}{n} is coprime to 105. This is equivalent to asking whether there are infinitely many integers which only have the digits 0,1 in base 3, 0,1,2 in base 5, and 0,1,2,3 in base 7. In general, one can ask whether there are infinitely many integers which only have 'small' digits in multiple bases simultaneously. For two bases this was established in 1975 by Erdos, Graham, Ruzsa, and Straus, but the case of three or more bases is much more mysterious. I will discuss recent joint work with Ernie Croot, in which we prove that (assuming the bases are sufficiently large) there are infinitely many integers such that almost all of the digits are small in all bases simultaneously.