Simultaneous diophantine approximation in the real and p-adic fields with nonmonotonic error function*

In this paper, we show that if the volume sum ( sumnolimits_{h = 1}^infty {{h^{n - 1}}{Psi^t}(h)} ) converges for a function ψ (not necessarily monotonic), then the set of points ( left( {x,{w_1}, ldots, {w_{t - 1}}} right) in {mathbb R} times {{mathbb Q}_{{p_1}}} times ldots times {{mathbb Q}_{{p_{t - 1}}}} ) that simultaneously satisfy the inequalities ( left| {P(x)} right| < Psi (H) {text{and}} {left| {Pleft( {{w_i}} right)} right|_{{p_i}}} < Phi (H), 1 leqslant i leqslant t - 1 ), for infinitely many integer polynomials P has measure zero.