Summary: Let \(B\) be the set of natural numbers which can be represented as a sum of two squares of integers. The main result of this paper is a lower sieve estimate contained~in Theorem 1. Let \(c\) be a non-zero integer and \(a,b \in \mathbb N\) such that \((a,b) = 1\) and \((ab,2c) = 1\). Further, let \[ S(x):= \sharp \{n:n \leq x,\, a(n+c) = b(m+c), \, (a,n+c)=1,\, n,m \in B \}. \] Then there exists a positive constant \(\vartheta = \vartheta (a,b,c)\) such that \[ S(x) \geq \vartheta \frac{x}{\log x} \] for \(x \geq x_0 = x_0(a.b.c)\). As a consequence we show that every natural number can be represented infinitely often as a quotient of two finite products of elements of \(B + c\). In the case \(c=1\) we prove (see Theorem~4) that, if \( a \in \mathbb N\), \(a = 2^r b\) with \((2,b)=1\), there exist infinitely many representations \[ a = \prod_{i=1}^{s} (n_i +1)^{\varepsilon_i}, \quad\;\varepsilon_i \in \{-1,1\},\, n_i \in B\;(i=1,\ldots,s) \] such that, if \(0 \leq r \leq 1\), then \(s=2\) and, if \(r \geq 2\), then \(s=r+1\).