On a new construction in group theory

This paper continues the investigation of the groups RF(G)\mathcal{RF}(G) introduced and studied in [I.M. Chiswell and T.W. Müller, A class of groups with canonical ℝ-tree action, Springer LNM, to appear]. Two new concepts, that of a test function, and that of a pair of locally incompatible (test) functions are introduced, and their theory is developed. As application, we obtain a number of new quantitative as well as structural results concerning RF(G)\mathcal{RF}(G) and its quotient RF(G)/E(G)\mathcal{RF}(G)/E(G) modulo the subgroup E(G) generated by the elliptic elements. Among other things, the cardinality of RF(G)\mathcal{RF}(G) is determined, and it is shown that both RF(G)\mathcal{RF}(G) and RF(G)/E(G)\mathcal{RF}(G)/E(G) contain large free subgroups, and that their abelianizations both contain a large ℚ-vector space as direct summand.     

Normal growth of large groups, II

We show that a large class of groups has normal subgroup growth of type n^{log n}, which is the same as the growth type of a free group of rank 2.Received: 21 June 2004     

Weakly multiplicative arithmetic functions and the normal growth of groups

Archiv der Mathematik - We show that an arithmetic function which satisfies some weak multiplicativity properties and in addition has a non-decreasing or $$\log $$ -uniformly continuous normal...     

The order of elements in Sylow p-subgroups of the symmetric group

Define a random variable ξ_n by choosing a conjugacy class C of the Sylow p-subgroup of S_pn by random, and let ξ_n be the logarithm of the order of an element in C. We show that ξ_n has bounded variance and mean order log n /log p +O(1), which differs greatly from the average order of elements chosen with equal probability. This revised version was published online in June 2006 with corrections to the Cover Date.     

Sets with more differences than sums

We show that a random set of integers with density 0 has almost always more differences than sums.      

The irrationality of a number theoretical series

Denote by σ _k (n) the sum of the k-th powers of the divisors of n, and let . We prove that Schinzel’s conjecture H implies that S _k is irrational, and give an unconditional proof for the case k = 3. 2000 Mathematics Subject Classification Primary—11A25, 11N36, 11J72     

Sign changes of π (x,q,1) - π(x,q,a)

It is known that under the assumption of the generalized Riemann hypothesis the function π(x,q,1) - π(x,q,a) has infinitely many sign changes. In this article we give an upper bound for the least such sign change. Similarly, assuming the Riemann hypothesis we give a lower bound for the number of sign changes of π(x)-li x. The implied results for the least sign change are weaker than those obtained by numerical methods, however, our method makes no use of computations of zeros of the ζ-function. This revised version was published online in June 2006 with corrections to the Cover Date.     

Compactness and local compactness for {\mathbb{R}}-trees

We establish (geometric) criteria for an -tree to be compact and to be locally compact. It follows that locally compact -trees are separable. Received: 10 September 2007     

On Romanov’s constant

Mathematische Zeitschrift - We show that the lower density of integers representable as a sum of a prime and a power of two is at least 0.107. We also prove that the set of integers with exactly...     

The exponents in the prime decomposition of factorials

Let ν_p(n) be the exponent of p in the prime decomposition of n. We show that for different primes p, q satisfying some mild constraints the integers ν_p(n!) and ν_q(n!) cannot both be of a rather special form.