For a function $f$, it is known that $|f|_{U_2} \le |f|_{U_3} \le |f|_{U_4} \le\dots$

Is there an example for a function $f$ such that $|f|_{U_3} < |f|_{U_4}$ (i.e. they are not equal?). The bigger the gap between them, the better.

I think there is a known construction of Gowers doing it but have no reference.

  • $\begingroup$ I am not sure, but perhaps the tag [reference-request] fits here? $\endgroup$ – Asaf Karagila Jul 23 '11 at 11:10
  • $\begingroup$ Most certainly not [number-theory]. It should be some subset of analysis (experts should retag). If you don't get an answer here in a few days, you may want to ask at MathOverflow. Gowers sometimes reads and answers there, but I don't think he does here. $\endgroup$ – Willie Wong Jul 23 '11 at 12:42
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    $\begingroup$ Willie, this question arose from an advanced course on additive number theory. The tag fits. $\endgroup$ – Gadi A Jul 23 '11 at 12:54
  • $\begingroup$ @Theo: the definition of the Gowers uniform norms is not number theoretical in nature. Just because the $U^3$ norm is useful in number theory does not mean that the question is necessarily one about number theory. $\endgroup$ – Willie Wong Jul 23 '11 at 13:01
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    $\begingroup$ @Gadi: for completeness, you should also state the domain of your function (partly to justify your claim that this is about number theory). And for a better question on this site (since it is a general interest site), perhaps provide a definition of the Gowers norm or a link. $\endgroup$ – Willie Wong Jul 23 '11 at 13:03

Actually, I think any non-constant function would provide an example of strict inequality: the equalities you mention are proven by repeated application of the Cauchy-Schwarz inequality applied to a mixture of the function $f$ with the constant function 1, and equality in the conventional CS inequality would only hold if $f$ was also a constant: hence if not at some point in your proof the inequalities would become strict.

For the best possible gap, you should probably look at (in your case) cubic phase functions such as $e(x^3)$ - explicit calculation shows that this has large $U^4$ norm, but since it doesn't correlate well with any quadratic phase function it can't have a large $U^3$ norm (by the inverse $U^3$ result due to Green and Tao). The details of the calculation and how large the gap is would depend on which group you were taking the norms over.

EDIT: To be more precise, and to give a reference, the following is Exercise 11.1.12 in Tao and Vu:

If $P$ is a polynomial of degree $d$ with coefficients in $\mathbb{F}_p$ and $f(x)=e(P(x)/p)$ (defining $x\mapsto x/p$ so that it maps into $[0,1)$ in the obvious way) then we have $$\| f\|_{U^{d'}}=1$$ for all $d'>d$ but $$ \| f\|_{U^{d'}}\leq ((d-1)/p)^{1/2^d}$$ for all $1\leq d'\leq d$.

For instance, if $f(x)=e(x^3/p)$ then $ \| f\|_{U^{4}}=1$ but $ \| f\|_{U^{3}}\leq (2/p)^{1/8}$ which can be made arbitrarily small by increasing $p$.

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