We introduce a new arithmetic function $a(n)$ defined to be the number of random multiplications by residues modulo $n$ before the running product is congruent to zero modulo $n$. We give several formulas for computing the values of this function and analyze its asymptotic behavior. We find that it is closely related to $P_{1}(n)$, the largest prime divisor of $n$. In particular, $a(n)$ and $P_{1}(n)$ have the same average order asymptotically. Furthermore, the difference between the functions $a(n)$ and $P_{1}(n)$ is $o(1)$ as $n$ tends to infinity on a set with density approximately $0.623$. On the other hand, however, we see that (except on a set of density zero) the difference between $a(n)$ and $P_{1}(n)$ tends to infinity on the integers outside this set. Finally, we consider the asymptotic behavior of the difference between these two functions and find that $\sum _{n\leqslant x}(a(n)-P_{1}(n))\sim (1-\unicode[STIX]{x1D70B}/4)\sum _{n\leqslant x}P_{2}(n)$, where $P_{2}(n)$ is the second largest divisor of $n$.

Szemerédi's theorem states that a set of integers with positive upper density contains arbitrarily long arithmetic progressions. Bergelson and Leibman generalized this, showing that sets of integers with positive upper density contain arbitrarily long polynomial configurations; Szemerédi's theorem corresponds to the linear case of the polynomial theorem. We focus on the case farthest from the linear case, that of rationally independent polynomials. We derive results in ergodic theory and in combinatorics for rationally independent polynomials, showing that their behavior differs sharply from the general situation.

Algebraic higher-rank actions on connected groups are often remarkably rigid in their topological and measurable structure. In contrast to this, the author of this paper constructs uncountably many closed invariant sets and uncountably many invariant measures with positive entropy for irreducible algebraic ${\mathbb{Z}^d}$-actions on zero-dimensional groups.