The problem with using the norm (which would seem to be a very natural thing to do) is that the nonexistence of APs in a set provides a bound on the norm of the balanced function of that results, after the above argument, in a weaker density increment than you get if you use a Fourier expansion: the norm is what you care about if you are looking for a density increment, so you lose information if you use the norm.

]]>Because the required bound on is signed, I have concluded that my proof is broken at the last point where I invoked the pigeonhole principle on . At that point it is necessary to choose a phase as is done in Tao &Vu. I think that the proof may be able to be salvaged, but the constant would not . That would still leave the core idea intact though.

]]>I have dug out a sketch of an alternative proof of Lemma 10.15 (Non-uniformity implies density increment) from Tao & Vu that uses the dual form of the Poisson summation formula (essentially taking averaging projections). Seeing as Lemma 10.15 is used when passing to a subspace, might it be possible to unpack my proof somehow and then replace by for the Bohr set case? The proof also improves the constant in Lemma 10.15 from 1/2 to , which makes me quite nervous that I have made a silly mistake. However, I am going to post the proof anyway in case something can be salvaged from the idea.

Lemma 10.15 + improved constant:

Write Z as a shorthand for . Let be a function with mean zero. Then there exists a codimension 1 subspace V of Z and a point , such that

Sketch of proof:

Choose such that .

Let W be the dimension 1 Fourier subspace generated by and let V be the orthogonal complement of . Let .

By the dual form of the Poisson Summation formula (Exercise 4.1.7 in Tao & Vu) .

From the Parseval identity and the fact that is real valued:

The result follows from the fact that is constant on codimension 1 affine subspaces and the pigeonhole principle.

]]>Regarding your very reasonable question, you are of course correct that there is no difference between and . I stupidly wrote when I was thinking of , and then copied the LaTeX for my mistake all over my blog post. Accordingly, I have updated to refer to , which is what I meant. So for clarity, I am proposing to compare against , obtaining frequency counts for possible values of at each density level-set in the power set of .

I don’t think that I have explained very well why I find the experiment interesting. Certainly, I can update my rather rushed blog post with a clearer explanation. However, even if I explain that clearly, I am far from confident that the experimental data would be of wider interest. If the experiment *is* of interest, then I can move material to this blog or to the wiki. On the other hand, if the experiment is not of interest, I still feel honour bound to complete the experiment before moving on.

Could I ask what the difference is between and in what you write?

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