Let \(h \ge 2\) , and let \(\textbf{b} = (b_1,\dots ,b_h)\in \mathbb {Z}^h\) be a zero-sum vector with nonzero coordinates. For a set \(A=\{a_1<a_2<\cdots \}\subseteq \mathbb {N}\) , let \(r_{A,\textbf{b}}(n)\) denote the number of h-tuples \((x_1,\ldots ,x_h)\) of pairwise distinct elements of A satisfying \(b_1x_1+\cdots +b_hx_h=n\) . We study density restrictions on sets A for which these representation counts remain small, obtaining analogues of the classical density theorem for infinite Sidon sets. In the case \(\textbf{b} = (c_1,-c_1,\dots ,c_k,-c_k)\) , we prove that if \(A(x)/(x/\log x)^{1/2k}\rightarrow \infty \) , then \(\frac{1}{x}\sum _{|n|\le x} r_{A,\textbf{b}}(n)\rightarrow \infty \) , whereas if \(A(x)\gg x^{1/2k}\) , then \(\frac{1}{x}\sum _{|n|\le x} r_{A,\textbf{b}}(n)\gg \log x\) . This recovers Chen’s theorem on \(B_{2k}\) -sequences. For general zero-sum vectors \(\textbf{b}\) , we prove analogous bounds under gap conditions: if \(a_{n+1}-a_n=o(n^{h-1}\log n)\) , then \(\frac{1}{x}\sum _{|n|\le x} r_{A,\textbf{b}}(n)\rightarrow \infty \) , whereas if \(a_{n+1}-a_n\ll n^{h-1}\) , then \(\frac{1}{x}\sum _{|n|\le x} r_{A,\textbf{b}}(n)\gg \log x\) .
