Its Polymath time. My resources, especially time and most importantly skill, are limited and therefore I have to restrict myself a little. Let me just sketch where I set the boundaries, what I want to try and what a possible (successful) outcome might look like. Some acquaintance with Tim Gowers’s proof of Roth’s theorem and its (hoped for) connection to EDP is necessary.
What is the idea?
Let me first collect some observations:
- ‘Translation’ acts as a group on APs and is periodic on APs with common difference.
- ROI starts with some representation of the translation group in terms of rank one projections using exponentials.
- An elementary formula from Fourier Analysis describing the interaction of translation and exponentials is used to express the exponentials in terms of Fourier coefficients of some characteristic function and its translates.
- The result is an ‘efficient’ representation, i.e. it allows to deduce unbounded discrepancy.
Translation as described above lives on the domain of the Fourier coefficients. We do not lose information if we consider it for the corresponding Fourier series. On all reasonable spaces translations form a strongly continuous (semi-) group of linear operators. For periodic strongly continuous groups we have representations of the group. If we work with rotation groups on with we even have a tensorial representation in terms of exponentials. By the way, if we choose our space carefully, exponentials are at least approximative eigenvectors of translations (rotations).
The idea now is to get information on discrepancy on (the domain of the Fourier coefficients) by studying ROI on various spaces of Fourier series, like or .
If I am not able to translate the ROI idea to the infinite section directly I will also try to use the so called ‘Complex Inversion Formula’, expressing the integral over the group in terms of some inverse Laplace transform. This can be seen as an infinite dimensional version of Perron’s formula. However that would be a ‘last try’ since it is not connected to Polymath anymore.
What is the goal?
The result I am aiming for looks roughly as follows:
Let be a Banach (or better a Hilbert) space and let be a strongly continuous (maybe only semi-)group of linear operators (translations, rotations, periodic?) with generator . If the resolvent satisfies some conditions (including norm estimates) then ‘something’ has unbounded ‘discrepancy’.
That sounds incredibly naive, since Tim Gowers’s proof needed some clever estimates and preyed upon cancellations for different common differences . My hope is to hide these technicalities in an estimate on the resolvent. Such estimates like e.g. in the Hille-Yosida theorem are usually harder to obtain than to state. That is good news.
So far, that would be the first part. In a way this is just translating the main idea of the proof into some other language. The hard part would then be to apply the general theorem to other situations (maybe even EDP). Here the resolvent has to be estimated and technicalities enter. However, I do not want to think that far ahead.