On tight sets of hyperbolic quadrics

We prove that the parameter $x$ of a tight set $\mathcal{T}$ of a hyperbolic quadric ${\mathsf{Q}}^{+}(2n+1,q)$ of an odd rank $n+1$ satisfies ${x\choose 2}+w(w-x)\equiv 0\mod q+1$,where$w$isthenumberofpointsof$$\mathcal{T}$$inanygeneratorof$$\mathsf{Q}$^+(2n+1,q)$. As this modular equation should have an integer solution in $w$ if such a $\mathcal{T}$ exists, this condition rules out roughly at least one half of all possible parameters $x$. It generalizes a previous result by the author and K. Metsch shown for tight sets of a hyperbolic quadric ${\mathsf{Q}}^{+}(5,q)$ (also known as Cameron-Liebler line classes in $\mathrm{P}\mathrm{G}(3,q)$).