Fractional parts of generalized polynomials at prime arguments (Q6614346)

Let \(f\) be a monic polynomial with real coefficients. Consider a pair \((\alpha,\beta)\) such that there exists \(t\geq 1\) satisfying the condition \N\[\N\max(1,|m|)^{t+\varepsilon}\max(1,|n|)^{t+\varepsilon}||m\alpha+n\alpha\beta||\gg_{\varepsilon} 1\N\]\Nfor any \(\varepsilon>0\) and any \((m,n)\in\mathbb{Z}^2\), \((m,n)\neq (0,0)\). Let \(\{p_n\}\) be a sequence of all prime numbers.\N\NThen there exists effectively computable \(\theta\) (depending on \(t\) and degree of \(f\) such that the discrepancy \(D_N\) of the sequence \(\{[f(p_n)\alpha]\beta\}_{2\leq p_N\leq N}\) can be bounded from above as \N\[\ND_N\ll_{\alpha,\beta,\varepsilon} N^{-\theta+\varepsilon}.\N\]\NSimilar upper bound also holds for the quantity \(\min_{2\leq p_n\leq N} ||[f(p_n)\alpha]\beta||\).