We combine the theory of Bailey chains and the theory of binary quadratic forms to show that there are large classes of q-series which are not theta series but whose coefficients are almost all 0. We interpret some examples in terms of simple partition functions.

By using Riemann–Hilbert problem based techniques, we obtain the asymptotic expansion of lacunary Toeplitz determinants detN [ cla−mb [ f ] ] generated by holomorhpic symbols, where la = a (resp. mb = b) except for a finite subset of indices a = h1, . . . , hn (resp. b = t1, . . . , tr). In addition to the usual Szegö asymptotics, our answer involves a determinant of size n + r.

Let {an}^_ao be a sequence which satisfies a linear recurrence of order k +1. We are herein concerned with the lacunary subsequences {®mn+b}Z=-ao> where m and b are fixed integers, so called because they consist of the terms from {aJ with lacunae, or gaps, of length m between them. In [5], [2], and [3] it has been shown that, for any m and A, the subsequences {amn+b} also satisfy a linear recur...

We give a simple argument proving the lonely runner conjecture in the case where the speed of the runners forms a certain lacunary sequence. This improves an earlier result by Pandey, and is then used to derive that for each number of runners the lonely runner conjecture is true for a set of nonzero measure in a natural probability space.