If there are some of norm strictly between 0 and 1, then would be strictly less than , and the above string of inequalities would give , which is absurd.

]]>*[Updated, thanks – T.]*

Well, there is a broad theme in several areas of analysis that an inverse theorem (that asserts the largeness of some norm or norm-like quantity implies the presence of a specific structure) implies a structure theorem (that asserts that arbitrary objects can be decomposed into a superposition of specific structures, plus an error of small norm). In the case of the concentration compactness result above, the inverse theorem is quite trivial: a function with large norm contains a large component concentrated at a point. But one could similarly start with, say, the inverse theorem that says that a function with large Gowers norm correlates with a Fourier character, and obtain an analogous concentration compactness result that would assert that a sequence of bounded (in ) functions could be resolved into a superposition of frequency modulated profiles, plus an error with small norm; this is close to certain “arithmetic regularity lemmas” that are already used in the additive combinatorics literature.

]]>i wonder is this idea of profile decomposition related to fourier linear bias?

if so, could you say something more about it. thanks

reader

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