Growth-rate conditions for power-series coefficients

Determine conditions on the exponents a_n that guarantee exponential, subexponential, or otherwise specified growth rates for the coefficients r(n) of the associated q-product.

Background

The q-factorization relates the exponent sequence a_n to the coefficient sequence r(n), but the paper does not characterize how restrictions on a_n control the asymptotic growth of r(n). The question specifically asks for hypotheses yielding exponential, subexponential, or prescribed rates of growth.