LLL: a tool for effective diophantine approximation

International audienceThe purpose of this paper is to survey in a unified setting some of the results in diophantine approximation that the LLL algorithm can make effective in an efficient way. We mostly study the problems of finding good rational approximations to vectors of real and p-adic numbers, and of finding approximate linear relations between vectors of real numbers. We also discuss classical applications of those eective versions, among which Mertens' conjecture and the effective solution of diophantine equations