On the measure of ``major arcs'' in the Farey partition (Q2857867)

Let \(1\leq Q\leq \sqrt{X}.\) Let \(a\) and \(q\) be integers such that \(1\leq q\leq Q,\, (a,q)=1.\) Let \(I_{a,q}\) be the interval of the form NEWLINE\[NEWLINEI_{a,q}=\left(\frac{a}{q}-\frac{Q}{qX},\frac{a}{q}+\frac{Q}{qX}\right).NEWLINE\]NEWLINE Let \(|k|\ll\log X\) be an integer. Let \(\beta\) be an irrational, algebraic number and \(t\) be a real number, such that \( X^{-1/3}\leq|t|\leq1/2,\quad1<\beta<2. \) Let NEWLINE\[NEWLINE T=\left\{ t\,\Bigl| \,\exists\, a_1,q_1,a_2,q_2: t\beta\in I_{a_1,q_1},\, t+k\in I_{a_2,q_2} \right\}. NEWLINE\]NEWLINE It is proved in the paper that for any \(\varepsilon>0\) one has NEWLINE\[NEWLINE{mes}\left(T\right)\ll_{\varepsilon}X^{-2+\varepsilon}(Q_1Q_2)^2.NEWLINE\]