On the Symmetry Integral

We give a level one result for the “symmetry integral”, say $I_f(N,h)$, of essentially bounded $f:\mathbb{N}\to \mathbb{R}$; i.e., we get a kind of “square-root cancellation” \thinspace bound for the mean-square (in $N<x\le 2N$) of the “symmetry” \thinspace of, say, the arithmetic function $f:=g\ast \text{1}$, where $g:\mathbb{N}\to \mathbb{R}$ is such that $\mathrm{\forall }ϵ>0$ we have $g(n)\ll_{\epsilon} n^{\epsilon}$,andsupportedin$[1,Q]$,with$Q\ll N$(so,theexponentof$Q$relativeto$N$,saythelevel$\lambda:=(\log Q)/(\log N)$is$\lambda < 1$), where the symmetry sum weights the $f-$values in (almost all, i.e. all but $o(N)$ possible exceptions) the short intervals $[x-h,x+h]$ (with positive/negative sign at the right/left of $x$), with mild restrictions on $h$ (say, $h\to \mathrm{\infty }$ and $h=o(\sqrt{N})$, as $N\to \mathrm{\infty }$).