Independent Linear Forms on the Group $$\varOmega _p$$Ωp

Let $$\varOmega _p$$Ωp be the group of p-adic numbers, and $$ \xi _1$$ξ1, $$\xi _2$$ξ2, $$\xi _3$$ξ3 be independent random variables with values in $$\varOmega _p$$Ωp and distributions $$\mu _1$$μ1, $$\mu _2$$μ2, $$\mu _3$$μ3. Let $$\alpha _j, \beta _j, \gamma _j$$αj,βj,γj be topological automorphisms of $$\varOmega _p$$Ωp. We consider linear forms $$L_1 = \alpha _1\xi _1 + \alpha _2 \xi _2+ \alpha _3 \xi _3$$L1=α1ξ1+α2ξ2+α3ξ3, $$L_2=\beta _1\xi _1 + \beta _2 \xi _2+ \beta _3 \xi _3$$L2=β1ξ1+β2ξ2+β3ξ3 and $$L_3=\gamma _1\xi _1 + \gamma _2 \xi _2+ \gamma _3 \xi _3$$L3=γ1ξ1+γ2ξ2+γ3ξ3. We describe all coefficients of the linear forms for which the independence of $$L_1$$L1, $$L_2$$L2 and $$L_3$$L3 implies that distributions $$\mu _1$$μ1, $$\mu _2$$μ2, $$\mu _3$$μ3 are idempotent. This theorem is an analogue of the well-known Skitovich–Darmois theorem, where a Gaussian distribution on the real line is characterized by the independence of two linear forms.