Bounded asymptotic bases for linear forms

For a vector of positive integers $\mathbf{b} = (b_1,\ldots,b_h)$ with $\gcd(b_1,\ldots,b_h) = 1$, we study sets $A \subseteq \mathbb{N}$ for which every sufficiently large integer has a bounded positive number of representations \[ n = b_1 x_1 + \cdots + b_h x_h \qquad (x_1,\ldots,x_h\in A). \] We