One problem that has bugged me for some time (though I only seriously thought about it for a month several years ago) is to give a physical proof of the L^2 boundedness of Bourgain maximal function for averages along the squares.

To be precise, define the averages: for a function $f: \mathbb{Z} \to \mathbb{C}$, let $$ A_Nf(x) := N^{-1} \sum_{n=1}^N f(x-n^2) $$ and the maximal function $$ Mf(x) := \sup_{N \in \mathbb{N}} |A_Nf(x)| $$ for $x \in \mathbb{Z}$.

**Question**: Without resorting to the circle method, can one prove that $M: \ell^2(\mathbb{Z}) \to \ell^2(\mathbb{Z})$?

In particular, I suspect that there a physical proof analogous to the proofs for $L^2$ boundedness of Stein's spherical maximal function theorem as sketched out by Laba here: https://ilaba.wordpress.com/2009/05/23/bourgains-circular-maximal-theorem-an-exposition/ or as in Schlag's work: https://math.uchicago.edu/~schlag/papers/dukecircles.pdf In particular, I am happy with a physical proof of restricted weak-type $L^2$ boundedness.

My suspicion is that Bourgain's proof, which uses the circle method, suggests that we should consider understand what happens on arithmetic progressions, decompose the characteristic function of a set into how arithmetic progressions and combine them in some way. However I could not see how to successfully deploy this strategy...