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We address the issue of variable selection in the regression model with veryhigh ambient dimension, that is, when the number of variables is very large.The main focus is on the situation where the number of relevant variables,called intrinsic dimension, is much smaller than the ambient dimension d.Without assuming any parametric form of the underlying regression function, weget tight conditions making it possible to consistently estimate the set ofrelevant variables. These conditions relate the intrinsic dimension to theambient dimension and to the sample size. The procedure that is provablyconsistent under these tight conditions is based on comparing quadraticfunctionals of the empirical Fourier coefficients with appropriately chosenthreshold values. The asymptotic analysis reveals the presence of two quitedifferent re gimes. The first regime is when the intrinsic dimension is fixed.In this case the situation in nonparametric regression is the same as in linearregression, that is, consistent variable selection is possible if and only iflog d is small compared to the sample size n. The picture is different in thesecond regime, that is, when the number of relevant variables denoted by stends to infinity as $n\to\infty$. Then we prove that consistent variableselection in nonparametric set-up is possible only if s+loglog d is smallcompared to log n. We apply these results to derive minimax separation ratesfor the problem of variable

Tight conditions for consistency of variable selection in the context of high dimensionality