  We address the issue of variable selection in the regression model with very
high ambient dimension, i.e., when the number of covariates is very large. The
main focus is on the situation where the number of relevant covariates, called
intrinsic dimension, is much smaller than the ambient dimension. Without
assuming any parametric form of the underlying regression function, we get
tight conditions making it possible to consistently estimate the set of
relevant variables. These conditions relate the intrinsic dimension to the
ambient dimension and to the sample size. The procedure that is provably
consistent under these tight conditions is simple and is based on comparing the
empirical Fourier coefficients with an appropriately chosen threshold value.
