Gowers norms for singular measures

Publication date: 12 August 2013

Abstract: Gowers introduced the notion of uniformity norm

| f |_{U^{k} (G)}

of a bounded function

f : G i g h t a r r o w m a t h b b R

on an abelian group

G

in order to provide a Fourier-theoretic proof of Szemeredi's Theorem, that is, that a subset of the integers of positive upper density contains arbitrarily long arithmetic progressions. Since then, Gowers norms have found a number of other uses, both within and outside of Additive Combinatorics. The

U^{k}

norm is defined in terms of an operator

r i a n g l e^{k} : L^{i n f t y} (G) m a p s t o L^{i n f t y} (G^{k + 1})

. In this paper, we introduce an analogue of the object

r i a n g l e^{k} f

when

f

is a singular measure on the torus

m a t h b b T^{d}

, and similarly an object

| m u |_{U^{k}}

. We provide criteria for

r i a n g l e^{k} m u

to exist, which turns out to be equivalent to finiteness of

| | m u | |_{U^{k}}

, and show that when

m u

is absolutely continuous with density

f

, then the objects which we have introduced are reduced to the standard

r i a n g l e^{k} f

and

| f |_{U^{k} (m a t h b b T)}

. We further introduce a higher-order inner product between measures of finite

U^{k}

norm and prove a Gowers-Cauchy-Schwarz inequality for this inner product.

Mathematics Subject Classification ID

Function spaces arising in harmonic analysis (42B35) Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type (42B10) Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type (42A38) Multipliers in one variable harmonic analysis (42A45) Arithmetic progressions (11B25) Hausdorff and packing measures (28A78) Trigonometric series of special types (positive coefficients, monotonic coefficients, etc.) (42A32)

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