Coordinate sum and difference sets of \(d\)-dimensional modular hyperbolas (Q2855609)

The authors of the present paper investigate subsets of modular hyperbolas that are sum- or difference-dominant. In particular, they estimate the sum- to difference-dominance ratio.NEWLINENEWLINELet \(d\) be a positive integer. The \(d\)-dimensional modular hyperbola \(H_d(a,n)\) is defined by NEWLINE\[NEWLINEH_d(a;n):=\{(x_1,\ldots,x_d)\in\{1,\ldots,n-1\}^d: x_1\cdots x_d\equiv a\pmod n\},NEWLINE\]NEWLINE where \((a,n)=1\). Furthermore let \(\overline{S}_d(m;a;n)\) be the generalized signed sumset, i.e., NEWLINE\[NEWLINE\overline{S}_d(m;a;n)=\{x_1+\cdots+x_m-\cdots-x_d\pmod n: (x_1,\ldots,x_d)\in H_d(a;n)\},NEWLINE\]NEWLINE where \(m\) is the number of plus signs. The sumset to difference sets ratio \(c_2(a;n)\) is defined by NEWLINE\[NEWLINEc_2(a;n)=\#\overline{S}_2(2;a;n)/\#\overline{S}_2(1;a;n).NEWLINE\]NEWLINENEWLINENEWLINEAmong other results they show for the \(2\)-dimensional case that if \(N_k=\prod_{i=1}^kp_i\), where \(p_i\) is the \(i\){th} prime that is congruent to \(3\pmod 4\), and \(a\) is a fixed perfect square relatively prime to all the \(p_i\), then NEWLINE\[NEWLINEc_2(a;N_k)\asymp\log\log N_k,NEWLINE\]NEWLINE and for any \(t\geq2\), NEWLINE\[NEWLINEc_2(a;N_k^t)\asymp(\log\log N_k)^{-1}.NEWLINE\]NEWLINE For the case of higher dimension (\(d>2\)) they show that \(\#\overline{S}_d(m;a;n)=n\) whenever all prime factors of \(n\) exceed \(7\).