Let
. Define the average
of
over the square
integers by
We show
that
satisfies a local
scale-free
-improving
estimate, for
:
provided
is supported
in some interval of length
,
and
is the conjugate index. The inequality above fails for
. The maximal function
||
satisfies a similar sparse bound. Novel weighted and vector valued inequalities for
follow. A critical step in the proof requires the control of a logarithmic average over
of a function
counting the number of
square roots of
. One requires
an estimate uniform in
.
Keywords
improving discrete quadratic residues, sparse bounds,
circle method