According to the Heyde theorem the Gaussian distribution on the real line is characterized by the symmetry of the conditional distribution of one linear form in independent random variables given another. We prove an analogue of this theorem for linear forms in two independent random variables with values in an a-adic solenoind without elements of order 2, assuming that the characteristic functions of the random variables do not vanish, and coefficients of the linear forms are topological automorphisms of the a-adic solenoid.

中文翻译：

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关于a-adic Solenoids的表征​​定理

根据海德定理，实线上的高斯分布的特征在于给定另一种独立随机变量中一种线性形式的条件分布的对称性。假设随机变量的特征函数不消失，并且线性形式的系数是拓扑的，我们证明了该定理的类比，用于两个独立随机变量中的线性形式，其值在 a-adic 电磁线圈中，没有 2 阶元素a-adic 螺线管的自同构。