This paper is devoted to the characterization of the lack of compactness of the Sobolev embedding of H N (R 2N ) into the Orlicz space using Fourier analysis. The approach adopted in this paper is strikingly different from the one used in 2D in [4, 6, 7] , which consists in tracking the large values of the sequences considered. The analysis we employ in this work is inspired by the strategy of P. Gérard in [13] and is based on the notion introduced in [8] of being log-oscillating with respect to a scale.