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Abstract. A well-known characterization of quasicircles is the following: A Jordan curve J in the complex plane is a quasicircle if and only if it is the image of the unit circle under a quasisymmetric embedding. In this paper we try to characterize a subclass of quasicircles, namely, symmetric quasicircles, by introducing the concept of asymptotically symmetric embeddings. We show that a Jordan curve J in the complex plane is a symmetric quasicircle if and only if it is the image of the unit circle under an asymptotically symmetric embedding.

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In this article, we estimate the average number of real zeros of a class of trigonometric polynomials of the form T = n k=1 a k b k cos k where the a k 's are independent standard normally distributed random variables and the b k 's are binomial coefficients n k 1 2 .