Let \(L\) be a lattice in \(\mathbb R^ n\) and let \(L^*\) be its dual. The author shows that for each \(x\in\mathbb R^ n\setminus L\) there exists a nonzero \(v\in L^*\) such that \[ \frac{| \{(x,v)\}|}{\| v\|}\geq c_ n\cdot d(x,L), \] where \((x,v)\) is the usual inner product on \(\mathbb R^ n,\) \(\{\alpha\}\) the minimal distance of \(\alpha\) to an integer, \(d(x,L)\) is the distance from \(x\) to \(L\) and \(c_ n\geq (6n^ 2+1)^{-1}.\) The proof is not constructible. The best known constructible proof gives a value \(c_ n\geq 9^{-n}.\)