On a multiplicative hybrid problem (Q1085203)

Let \(A\) and \(B\) be subsets of \(\{N,N+1,...,2N\}\) with \(| A| \gg N\) and \(| B| \gg N\). Then it is proved that there exist integers \(a,b,c\) with \(a\in A\), \(b\in B\) and \(| ab-c^ 2| \ll (c \log c)^{1/2}\). More general results are also proved. These are ``hybrid multiplicative analogues'' of ``hybrid additive problems'' which were studied earlier by A. Balog, the second author, and C. L. Stewart.