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You are asking for which functions $f$ the sequence $f(n)$ is equidistributed modulo 1. This is a whole area of mathematics, which began with the work of Weyl in 1916, who discovered the connection be …

Pick a sequence of real numbers $x_i$ as follows. Put $x_0=1$. If $x_i$ is chosen, then pick $x_{i+1}\in[0, x_i]$ according to the uniform distribution. Obviously we have $x_i\rightarrow 0$ with proba …

It depends a lot what you exactly you mean by "constructive". You can replace randomization by a greedy algorithm, or by a deterministic strategy similar to the one developed by Beck for combinatorial …

Let $f$ and $g$ be random mappings. If they commute, then $f(g(1))=g(f(1))$, and this happens with probability $n^{-1}$. Now $f(g(2))=g(f(2))$ also holds with probability $n^{-1}$, but these events ne …

The average score difference does not suffice to predict the probability of the outcome. Suppose all players in team A are of equal strength, while all but one player in team B are somewhat stronger t …

A similar problem, called the Coupon Collector's younger brothers, has been studied by Foata, Han and Lass (Séminaire Lotharingien de Combinatoire, B47a, 20 pages, 2001, obtainable via http://math.uni …

Consider the set of $n\times n$-matrices with entries in $\{0,1\}$ which have at most $r$ distinct rows. The number of such matrices is $2^{rn}r^n$. As long as $n$ and $n-r$ tend to infinity, we have …

Which formula to prefer depends mainly on what you want to do with it. Do you need high precision, or do you have to do complicated things with the approximation? The expansion in Hermite-polynomials …

The probability that a fixed entry of $v$ is 1 equals $2^{-n^{s+t}}$. Hence the expected Hamming weight of $v$ is $2^{n-n^{s+t}}$. If $s+t\geq 1$, this implies that with high probability the Hamming w …