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Ahern and Rudin have given an explicit construction of a totally realembedding of $S^3$ in $\mathbb{C}^3$. As a generalization of their example, wegive an explicit example of a CR regular embedding of $S^{4n-1}$ in$\mathbb{C}^{2n+1}$. Consequently, we show that the odd dimensional sphere$S^{2m-1}$ with $m>1$ admits a CR regular embedding in $\mathbb{C}^{m+1}$ ifand only if $m$ is even.

CR regular embeddings of $S^{4n-1}$ in $\mathbb{C}^{2n+1}$