The Littlewood-Gowers problem |
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Authors: | T Sanders |
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Institution: | (1) Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge, CB3 0WA, England |
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Abstract: | The paper has two main parts. To begin with, suppose that G is a compact abelian group. Chang’s Theorem can be viewed as a structural refinement of Bessel’s inequality for functions
ƒ ∈ L
2(G). We prove an analogous result for functions ƒ ∈ A(G), where A(G) is the space
endowed with the norm
, and generalize this to the approximate Fourier transform on Bohr sets.
As an application of the first part of the paper, we improve a recent result of Green and Konyagin. Suppose that p is a prime number and A ⊂ ℤ/pℤ has density bounded away from 0 and 1 by an absolute constant. Green and Konyagin have shown that ‖χ
A
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A(ℤ/pℤ) ≫ ɛ (log p)1/3−ɛ; we improve this to ‖χ
A
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A(ℤ/pℤ) ≫ ɛ (log p)1/2−ɛ. To put this in context, it is easy to see that if A is an arithmetic progression, then ‖χ
A
‖
A(ℤ/pℤ) ≪ log p. |
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Keywords: | |
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