Some new results in probabilistic group theory (Q1249232)

Let \(G\) be an Abelian group of \(n\) elements. Assume that for each fixed \(l\) the number of elements of order \(l\) is \(o(n)\) as \(n\to\infty\). Let \(k=\frac{\log n}{\log 2}+0(1)\). Choose \(k\) elements of \(G\) at random. Let these elements be \(g_1,\dots,g_k\) and denote by \(R(g)\) the number of solutions of \(g=\sum_{i=1}^k\varepsilon_ig_i,\varepsilon_i = 0 \text{ or } 1\). Denote finally by \(d(r)\) the number of elements of \(G\) with \(R(g) = r\). The authors prove (among others) that of reach fixed \(r d(r)=(1+0(1))n e^{-lambda}\lambda=\frac{2^k}n\), with probability tending to \(1\) as \(n\to\infty\). Several applications and unsolved problems are discussed.