Asymptotic behaviour of the partition function

Given a pair of positive integers $m$ and $d$ such that $2\leqslant m\leqslant d$, for integer $n\geqslant 0$ the quantity $b_{m,d}(n)$, called the partition function is considered; this by definition is equal to the cardinality of the set
$$ \{(a_0,a_1,…):n=\sum_ka_km^k, a_k\in\{0,…,d-1\}, k\geqslant 0\}. $$
The properties of $b_{m,d}(n)$ and its asymptotic behaviour as $n\to\infty$ are studied. A geometric approach to this problem is put forward. It is shown that
$$ C_1n^{\lambda_1}\leqslant b_{m,d}(n)\leqslant C_2n^{\lambda_2}, $$
for sufficiently large $n$, where $C_1$ and $C_2$ are positive constants depending on $m$ and $d$, and $\lambda_1=\varliminf\limits_{n\to\infty}\dfrac{\log b(n)}{\log n}$ and $\lambda_2=\varlimsup\limits_{n\to\infty}\dfrac{\log b(n)}{\log n}$ are characteristics of the exponential growth of the partition function. For some pair $(m,d)$ the exponents $\lambda_1$ and $\lambda_2$ 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 $C_1$ and $C_2$ are obtained.