On the Symmetry Integral

Abstract

We give a level one result for the "symmetry integral", say $I_f(N,h)$, of essentially bounded $f:\N \to \R$; i.e., we get a kind of "square-root cancellation" \thinspace bound for the mean-square (in $N 0$ we have $g(n)\ll_ε n^ε$, and supported in $[1,Q]$, with $Q\ll N$ (so, the exponent of $Q$ relative to $N$, say the level $λ:=(\log Q)/(\log N)$ is $λ< 1$), 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, $h\to \infty$ and $h=o(\sqrt N)$, as $N\to \infty$).