Hi. Is anything very precise known about the almost everywhere
convergence of sums of the sort

\sum_{n=1}^N a_n f(n*\theta), .........(1)

where f is function of period 1 in L^2[0,1) with f^(0) = 0 and {a_n}
square-summable (say)? (If you don't see why sums of this sort would
be convergent, think about them either in terms of Carleson's
theorem, or in terms of the ergodic theorem.)

The strongest result I'm aware of is the rather trivial result of
Gaposhkin that the series converges when f has an absolutely
convergent Fourier series. (I haven't been able to locate Gaposhkin's
paper, but I say trivial because the result can be, and probably was,
proven using an old technique of Merten's about double summation when
one of the sums is absolutely convergent.)

This paper (http://130.44.194.100/tran/1997-349-10/S0002-9947-97-
01837-0/S0002-9947-97-01837-0.pdf#search=%22On%20the%20convergence%
20of%20and%20the%20LIP%201%2F2%22) proves that the same result cannot
be extended to any Lipschitz class below those associated with
absolutely convergent Fourier series (i.e. below Lip a where a >
1/2). I don't think this would rule out the following conjecture,
however:

If b_k are the Fourier coefficients of f, then (1) is convergent a.e.
whenever

\sum_{m=-infty}^\infty |\sum_{nk = m} a_n b_k|^2 ....(2)

converges.

Th convergence of this sum implies (in fact corresponds exactly to)
the convergence of (1) in L^2, so the result would certainly be
plausible in light of Carleson.

On the other hand, I can prove that if the conjecture is true, then
there is a square summable a_n so that \sum_{j,k} a_j a_k (j,k)^2/
(jk) diverges (where (j,k) indicate the greatest common divisor of
the two numbers). This sum has certain similarities to other
quadratic sums that make it seem likely converge always.

I think I have an approach that may work in proving my conjecture,
but I don't see any point in spending a great deal of time finding
out it doesn't work if the result is already known to be false.

Some other things that would help me:

* Information on the size of the growth of Fourier coefficients for
functions in Lip a for a less than or equal to 1/2. (e.g. for any
square summable c_n, is there a function f in Lip 1/2 so that |c_n| <
|f^(n)|?)

* Anything known about bounds for sums like (2). I can prove that
there are square summable a and b for which the sums don't converge,
but I think if a coefficient 1/d(m) or 1/d(m)^2 or some power of 1/log
(m) is put into the sum, it might converge. (Compare Hilbert's
inequality.)

* Is there a fairly readable proof of Carleson's theorem that
wouldn't take forever to get through? I've never seen a proof, and I
feel like it's cheating to work on problems like these.

Thanks for any suggestions. I'm an undergraduate at Purdue
University, and none of the faculty here really specialize in these
sort of questions, so any help is greatly appreciated. Sorry if this
was rather longer than most messages posted here.

Brad