Infinite Sidon-type sets for zero-sum linear forms

Let $h \geq 2$, and let $\mathbf{b} = (b_1,\dots,b_h)\in \mathbb{Z}^h$ be a zero-sum vector with nonzero coordinates. For a set $A=\{a_1<a_2<\cdots\}\subseteq\mathbb{N}$, let $r_{A,\mathbf{b}}(n)$ denote the number of $h$-tuples $(x_1,\ldots,x_h)$ of pairwise distinct elements of $A$ satisfying $b_1