We investigate the use of Büchi's techniques for Presburger arithmetic. More precisely, we show how to efficiently compute an automaton which accepts the set of solutions of a linear Diophantine equation (suitably encoded). Following Büchi, this gives a decision technique for the whole Presburger arithmetic. We show however how to compute more efficiently the automaton in the case of disequalities, inequalities and systems of linear Diophantine problems. We also show that such an “automaton algorithm” has a nearly optimal worst case complexity, both for the existential fragment and for the whole first-order theory.