Construct a probability measure $\mu$ on the circle by successive removal of middle third intervals with redistributions of the existing mass at the $n$th stage being determined by probability $p_n$ applied uniformly across that level. Assume that the sequence $\{p_n\}$ is bounded away from both $0$ and $1$. Then, for sufficiently large $N$, (estimates are given) the Lebesgue measure of any algebraic sum of Borel sets $E_1,E_2,\ldots,E_N$ exceeds the product of the corresponding $\mu(E_i)^\alpha$, where $\alpha$ is determined by $N$ and $\{p_n\}$. It is possible to replace 3 by any integer $M\geq 2$ and to work with distinct measures $\mu_1,\mu_2,\ldots,\mu_N$.

This substantially generalizes work of Williamson and the author (for powers of single-coin coin-tossing measures in the case $M=2$) and is motivated by the extension to $M=3$.

We give also a simple proof of a result of Yin and the author for random variables whose binary digits are determined by coin-tossing.

Employing the same technique as in our previous papers, we establish an intrinsic characterization of the direct product of a complex Euclidean ball and punctured planes in the category of Stein manifolds from the viewpoint of holomorphic automorphism group.

The shadowing lemma of Shantao Liao asserts that a “recurrent” quasi-hyperbolic string of a $C^1$-class diffeomorphism of a closed manifold might be closed up by a periodic orbit. In this note, we further show that the string can be closed exponentially up by the periodic orbit. Moreover, this statement also holds for a $C^1$-class map which is only locally diffeomorphic.

In this paper, we study the invariant theory of Viberg's $\Theta$-groups in classical cases. For a classical $\Theta$-group naturally contained in a general linear group, we show the restriction map, from the ring of invariants of the Lie algebra of the general linear group to that of the $\Theta$-representation defined by the $\Theta$-group, is surjective. As a consequence, we obtain explicitly algebraically independent generators of the ring of invariants of the $\Theta$-representation. We also give a description of the Weyl groups of the classical $\Theta$-groups.

We study homogeneous star products on cotangent bundles of Lie groups, we prove that Kontsevich star products are homogeneous and we characterize them by closed integral formulas.

We prove that there exist no simply connected homogeneous contact metric manifolds having nonpositive sectional curvature.

We first extend the results of Chan and Baruah on the modular equations of Ramanujan's cubic continued fraction $C(\tau)$ to all primes $p$ by finding the affine models of modular curves and then derive Kronecker's congruence relations for these modular equations. We further show that by its singular values we can generate ray class fields modulo 6 over imaginary quadratic fields and find their class polynomials after proving that $1/C(\tau)$ is an algebraic integer.