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An interesting article may be A sharp form of the Cramér-Wold theorem by Cuesta-Albertos, Fraiman and Ransford, which can be found here: klick Their Theorem 3.1 states that under a moment condition a measures is well-defined by the values of the Fourier-transform on a set which "is not contained in any projective hypersurface."


Let $\gamma(x) = (\alpha(x), \beta(x)),$ where $\alpha, \beta$ are the real and imaginary parts. A self-intersection corresponds to a simultaneous zero of $(\alpha(x)-\alpha(y))/(x-y)$ and $(\beta(x)-\beta(y)/(x-y),$ If you use the rational parametrization for the circle (the $\tan t/2$ trick), both expressions become polynomials, and the number of ...


Though your interests are partly separate from the purely mathematical framework here, the basic theory is well developed. Notation and terminology vary, of course: e.g., your "semi-ordering" is usually called a "partial ordering". In the case of the symmetric group $S_n$, a convenient modern treatment is given by Gordon James in The Representation Theory ...

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