The n-point correlations between values of a linear form

We show that the n-point correlation function for the fractional parts of a random Linear form in m variables has a limit distribution with power-like tail. The existence of the limit distribution follows from the mixing prop erty of flows on SL(m+1, R)/SL(m+1, Z). Moreover, we prove similar limit th eorems (i) for the probability to find the fractional part of a random line ar form close to zero and (ii) also for related trigonometric sums. For lar ge m, all of the above limit distributions approach the classical distribut ions for independent uniformly distributed random variables.