Finite groups with a small proportion of vanishing elements

Let 𝐺 be a finite group and let P v ⁢ ( G ) \mathrm{P}_{\mathbf{v}}(G) be the proportion of elements g ∈ G g\in G such that χ ⁢ ( g ) = 0 \chi(g)=0 for some irreducible character 𝜒. In a recent paper, we proved that if P v ⁢ ( G ) < P v ⁢ ( A 7 ) \mathrm{P}_{\mathbf{v}}(G)<\mathrm{P}_{\mathbf{v}}(A_{7}) , then P v ⁢ ( G ) = ( m − 1 ) / m \mathrm{P}_{\mathbf{v}}(G)=(m-1)/m for some 1 ≤ m ≤ 6 1\leq m\leq 6 . Here we classify all the finite groups 𝐺 such that P v ⁢ ( G ) = ( m − 1 ) / m \mathrm{P}_{\mathbf{v}}(G)=(m-1)/m and m = 1 , 2 , … , 6 m=1,2,\ldots,6 .