Sets whose differences avoid squares modulo

We prove that if ε ( m stretchy="false">) stretchy="false">→ 0 \varepsilon (m)\to 0 arbitrarily slowly, then for almost all alttext="m"> encoding="application/x-tex">m and any alttext="upper A subset-of double-struck upper Z Subscript m"> A ⊂<!-- ⊂ <mml:msub> Z encoding="application/x-tex">A\subset \mathbb {Z}_m such minus A"> −<!-- − encoding="application/x-tex">A-A does not contain non-zero quadratic residues we have alttext="StartAbsoluteValue EndAbsoluteValue less-than-or-slanted-equals Superscript 1 slash 2 epsilon Baseline period"> stretchy="false">| ⩽<!-- ⩽ <mml:msup> 1 / 2 . encoding="application/x-tex">|A|\leqslant m^{1/2-\varepsilon (m)}.