Fourier expansion is not needed for the converse direction. If , split up the summation into sums over arithmetic progressions modulo by making a change of variables .

]]>The contribution of has to be treated separately. If (1) holds for a sequence of functions that tend uniformly to a limit , then it is not hard to show that (1) also holds for .

]]>Write . Each natural number can be uniquely expressed as for some . This can be used to break up sums of the second type into sums of the first type. To go in the reverse direction, one needs to perform a Fourier expansion in .

]]>$$ 1 \leq H \leq \delta^{-C_d} N $$

be replaced with

$$ 1 \leq H \leq \delta^{C_d} N $$

instead?

*[Corrected, thanks – T.]*

*[Corrected, thanks – T.]*

Below remark 1:

“…if {\alpha} is rational in the sense that {m\alpha = 0} for some positive integer {m}…”

Do you mean that {m\alpha = 0 \mod 1}?

* is an element of a torus here, not a Euclidean space. -T]*