On the sum of bounded multiplicative functions over some special subsets of integers

Let $J_1,\dots,J_k \subseteq [0,1)$ be finite unions of intervals, $P_1(x), \dots, P_k(x) \in R[x]$ of degree at least one, $Q_{m_1, \dots, m_k}(x) = m_1P_1(x) + \ldots + m_k P_k(x)$, $m_1, \dots, m_k \in \mathbb Z$. Assume that $Q_{m_1, \dots, m_k}(x) - Q_{m_1, \dots, m_k}(0)$ has at least one irrational coefficient for every $(m_1, \dots, m_k) \not= (0, \dots, 0)$. Let $S:={n | n \in \mathbb{N}, {P_l(n)} \in J_l, l=1, \dots, k}$, $\lambda=$ Lebesgue measure. We shall prove the following theorem. Under the conditions stated above $$\sup_{g \in \mathcal{M}_1} |{{1}\over x} \sum_{n \leq x \atop n \in S} g(n) - {{\lambda(J_1) \dots (lambda(J_k)} \over x} \sum_{n \leq x}g(n)| =\tau_x \to 0$$ as $x\to infty$. Here $\mathcal{M}_1$ is the set of complex valued multiplicative functions $g$ satisfying $|g(n)| \leq 1 (n \in \mathbb{N})$.