Discrepancy estimates for some linear generalized monomials (Q2809368)

The authors study uniform distribution properties of sequences of the form \((\lfloor n \alpha \rfloor \beta)_{n \geq 1}\), subject to assumptions on the Diophantine approximation properties of \(\alpha\) and \(\beta\). The main result, Theorem 1.1, states the following: Let \(\alpha\) and \(\beta\) be real numbers such that \(1,\alpha\) and \(\alpha \beta\) are linearly independent over the rationals. Assume that both the pairs \((\alpha,\alpha \beta)\) and \((\beta,1/\alpha)\) are of finite type \(t \geq 1\), which means that for every \(\varepsilon>0\) there is \(c>0\) such that for all pairs \((m,n) \neq (0,0)\) of integers NEWLINE\[NEWLINE (\max(1,|m|))^{t+\varepsilon} (\max(1,|n|))^{t+\varepsilon} \|m \gamma + n \delta \| \geq c, NEWLINE\]NEWLINE where \((\gamma,\delta) = (\alpha,\alpha \beta)\) or \((\gamma,\delta) = (\beta,1/\alpha)\), respectively. Then NEWLINE\[NEWLINE D_N(\lfloor \alpha n \rfloor \beta) \ll _{\alpha,\beta,\varepsilon} N^{-1/(3t-2)+\varepsilon}. NEWLINE\]NEWLINE As a consequence, \(D_N(\lfloor \alpha n \rfloor \beta) \ll _{\alpha,\beta,\varepsilon} N^{-1+\varepsilon}\) for almost all \((\alpha,\beta)\). The final section of the paper contains three open problems concerning questions of this type.