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In the paper [G.Pantsulaia, On Uniformly Distributed Sequences on [-1/2,1/2],Georg. Inter. J. Sci. Tech.,4(3) (2013), 21--27], it was shown that$\mu$-almost every element of $\mathbf{R}^{\infty}$ is uniformly distributed in$[-\frac{1}{2}, \frac{1}{2}]$, where $\mu$ denotes Yamasaki-Kharazishvilimeasure in $\mathbf{R}^{\infty}$ for which$\mu([-\frac{1}{2},\frac{1}{2}]^{\infty})=1$. In the present paper the same setis studying from the point of view of shyness and it is demonstrated that it isshy in $\mathbf{R}^{\infty}$. In Solovay model, the set of all real valuedsequences uniformly distributed module 1 in $[-\frac{1}{2}, \frac{1}{2}]$ isstudied from the point of view of shyness and it is shown that it is theprevalent set in $\mathbf{R}^{\infty}$.

On structure of uniformly distributed sequences in $[-\frac{1}{2},\frac{1}{2}]$ from the point of view of shyness