In the section on the density increment argument, when you wrote “…conclude that the left-hand side of (1) can be expressed as…” I think you meant to refer to Equation (2) rather than Equation (1).

Thank you.

*[Corrected, thanks – T.]*

I think I (with some help from my teacher!) just figured out what the definition of should naturally be

where we define .

Sorry, I meant and .

]]>Secondly, I simply cannot wrap my head around what the definition of is. Usually, if I had to calculate such an integral I would try to approximate such a function by simple functions and then using the monotone convergence theorem or something similar. However, I do not quite know how to work with this integral of . Thank you for patiently reading my questions.

]]>Thanks for your quick reply, this is very helpful!

I missed that has another dependency on (coming from having atoms), hence I thought I can still have enough shifts independent of :(

]]>Almost periodicity will give a good contribution from , but only on a relatively small number of shifts (a lower bound of . The control of is not strong enough to control its contribution on such a sparse set of shifts, so one cannot conclude any nontrivial lower bound on the number of 3-term APs this way.

]]>It seems like after Proposition 9, we already found the right sigma algebra of Fourier measurable sets with complexity (say we start with being the trivial -algebra). We can then write

Almost periodicity should hold for as in your proof. By proposition 9, the residual would have small -norm, say . It seems like one can already conclude Roth’s theorem here.

Thank you!

]]>A function on the integers can be viewed as the restriction of a function on the reals.

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