O(n^{log log n}) learning algorithm for DNF under the uniform distribution

We show that a DNF with terms of size at most d can be approximated by a function with at most dO(d log 1/ε) non zero Fourier coefficients such that the expected error squared, with respect to the uniform distribution, is at most ε. This property is used to derive a learning algorithm for DNF, under the uniform distribution. The learning algorithm uses queries and learns, with respect to the uniform distribution, a DNF with terms of size at most d in time polynomial in n and dO(d log 1/ε). The interesting implications are for the case when ε is constant. In this case our algorithm learns a DNF with a polynomial number of terms in time nO(log log n), and a DNF with terms of size at most O(log n/log log n) in polynomial time.