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.

]]>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.

Thanks. I’m not very experienced with wikis.

Nets

]]>I created some sections on the wiki for your paper with Bateman and started to add some content based on the discussions we had last week. I don’t want this to dissuade anyone from writing different takes on whatever I have added, however. Indeed, I imagine having several different perspectives on things will be very useful. So please don’t worry about the stepping on of toes on my account.

I think you should be able to create an account on the wiki yourself; at least there is a link towards the top right of the site that gives that impression.

Olof

]]>I am interested in the project of getting useful physical space information from large Fourier coefficients.

If this can’t be done for one coefficient as in the cap-set question, perhaps can the presence of many large

coefficients help? What would constitute a counterexample convincing us that such an approach is futile?

One way I imagined being helpful at this stage was to write some notes helping to explain our argument. I see

you have a wiki for this, but I don’t have the appropriate permissions to modify it. I’ve noticed that

several of you plan on contributing to it and don’t want to step on your toes.

However let me know if an outline of the argument in section 6 and a description of some examples that necessitate

the difficulties might be helpful.

Nets

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