If \((A+A)/(A+A)\) is small, then the ratio set is large (Q2793760)

The classical sum-product problem is to set a good lower bound for \(\max\{|A+A|, |AA|\}\), where \(A\) is an arbitrary finite set of real or complex numbers. It is natural to investigate problems where more algebraic operations are involved. Let \(A+A\over {A+A}\) denote the set of \(a+b\over {c+d}\) numbers, where \(a,b,c,d\in A\) and \(c+d\not=0\). The paper shows that for a finite \(A\subset \mathbb{R}\), NEWLINE\[NEWLINE\Big|{A+A\over {A+A}}\Big|>> {|A|^{2+{2\over 25}} \over {|A:A|^{1\over 25}|\log |A|}},NEWLINE\]NEWLINE and as a corollary, if \(\big| {A+A\over {A+A}}\big|<<|A|^2 \), then \(|A:A|>>{ |A|^2 \over \log^{25}|A|}\).