Asymptotic behaviour of the partition function

Given a pair of positive integers m and d such that 2 ≤ m ≤ d, for integer n ≥ 0 the quantity bm,d(n), called the partition function is considered; this by definition is equal to the cardinality of the set (a0, a1, . . .) : n = Σkakmk, ak ∈ 0, . . . , d - 1, k ≥ 0. The properties of bm,d(n) and its asymptotic behaviour as n → ∞ are studied. A geometric approach to this problem is put forward. It is shown that C1nλ1 ≤ bm,d(n) ≤ C2nλ2 for sufficiently large n, where C1 and C2 are positive constants depending on m and d, and λ1 = lim/n→∞ log b(n)/log n and λ2 = limn→∞ log b(n)/log n are characteristics of the exponential growth of the partition function. For some pair (m, d) the exponents λ1 and λ2 are calculated as the logarithms of certain algebraic numbers; for other pairs the problem is reduced to finding the joint spectral radius of a suitable collection of finite-dimensional linear operators. Estimates of the growth exponents and the constants C1 and C2 are obtained.