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We prove local measure bounds on the tubular neighbourhood of the singularset of codimension one stationary integral $n$-varifolds $V$ in Riemannianmanifolds which have both: (i) finite index on their smoothly embedded part;and (ii) $\mathcal{H}^{n-1}$-null singular set. A direct consequence of such abound is a bound on the upper Minkowski content of the singular set of such avarifold in terms of its total mass and its index. Such a result improves onknown bounds, namely the corresponding bound on the $\mathcal{H}^{n-7}$-measureof the singular set established by A. Song ([10]), as well as the same boundsestablished by A. Naber and D. Valtorta ([6]) for codimension one areaminimising currents. Our results also provide more structural information onthe singular set for codimension one integral varifolds with finite index (onthe regular part) and no classical singularities established by N.Wickramasekera ([11]).

Quantitative Estimates on the Singular Set of Minimal Hypersurfaces with Bounded Index