Popular products and continued fractions (Q2200058)

Interest in the so-called ``sum-prduct problems'' originated in the paper by \textit{P. Erdős} and \textit{E. Szemerédi} [in: Studies in Pure Mathematics, Mem. of P. Turán, 213--218 (1983; Zbl 0526.10011)]. The new results are Theorem 1: Let \(A,B\hbox{ and }C\) be subsets of \(\mathbb{F}_p\). There is a constant \(b_0>1\), such that for all \(\varepsilon>0\), if \(\delta=\frac{1}{4}b_0^{-1/\varepsilon}\) and \(\min(|B|,|C|)\geq p^{\varepsilon}\), then for all sufficiently large \(p\) we have \[|A+B|+|A^{-1}+C|\gg \min(\sqrt{P|A|},|A|p^{\delta}|).\] In fact, if we write \(W=(A+B)\cup (A^{-1}+C)\), then we have \[|A|leq\frac{|W|^2}{p}+C_{\ast}|W|p^{-\delta(k)},\] where \(C_{\ast}\geq 6\) is an absolute constant and \(\delta(k)=2^{-(k+2)}\), where \[k\gg \log_{\min(|B|,|C|)} p.\] Theorem 3: There is an absolute constant \(\delta>0\) such that for all \(M,N\geq 2\) and all \(p\) sufficiently large, if \(B=\{1,\ldots,M\},\ C=\{1,\ldots,N\}\), and \(A\) is asubset of \(\mathbb{F}_p\), then either \[|A+B|+|A^{-1}+|C|\geq \sqrt{p|A|/2},\] or \[|A+B|+|A^{-1}+C|\geq C(p,M,N)\frac{|A|}{2}\lfloor\frac{\min(|B|,|C|)}{2}\rfloor^{\delta},\] where \[C(p,M,N)=C_0^{\log_p{\min (M,N)}},\hbox{ where }C_0>0\hbox{ is fixed}.\] Theorem 4: Given positive \(\varepsilon\) there exists \(M_0=M_0(\varepsilon)\) such that for all \(M\geq M_0\) one has \[|Z_M(p)|\ll_M p^{2w_M-1+\varepsilon (1-w_M)}.\] Notations: 1. The \textit{sum set\/}: \(A+B=\{a+b:\,a\in A,b\in B\}\). 2. The \textit{product set\/}: \(AB=\{ab:\,a\in A,b\in B\}\). 3. The Vinogradov symbols: \(f\ll g\Leftrightarrow \exists C>0\hbox{ with }f\leq Cg\),and \(f\gg g\hbox{ if and only if }g\ll f\). 4. For a prime \(p\): \(Z_M(p)=\{a\in \{1,\ldots, p-1\}:\,\max_{1\leq j\leq s(a)}\,b_j(a)\leq M\}\). Thus this is the set of all \(a\)'s in the given range for which all partial quotients of the continued fraction for \(a/p\) (i.e. \(a/p=[0;b_1,\ldots,b_s],\ b_s\geq 2\)) are bounded by \(M\). The layout of the paper is as follows: \S1. Introduction (\(6\) pages) -- introduces the sum-product problem, gives the results and historical remarks \S2. Bounds for rich linear fractional transformations (\(8\) pages) \S3. Proof of Theorem 3 (\(3\) pages) \S4. Proof of Theorem 4 (\(3 1/2\) pages) \S5. Proof of Theorem 5 (\(4 1/2\) pages) References (\(55\) items)