Bounded asymptotic bases for linear forms

For a vector of positive integers $\mathbf{b}=(b_1,\dots ,b_h)$ with $\gcd (b_1,\dots ,b_h)=1$, we study sets $A\subseteq \mathbb{N}$ for which every sufficiently large integer has a bounded positive number of representations $$n=b_1{x}_1+\cdots +b_h{x}_h(x_1,\dots ,x_h\in A)\mathrm{.}$$ We prove that such a set exists for every binary vector $\mathbf{b}\mathrm{\ne }(1,1)$, and for some general higher-dimensional families, including $\mathbf{b}=(u_1,p^d{u}_2,\dots ,p^{(h-1)d}{u}_h)$ where $p\nmid u_1\cdots u_h$.