We denote by ‖…‖ the distance to the nearest integer. Let α and β be real. W. M. Schmidt [5] proved that for ε > 0 and N>c1(ε) there is a natural number n such that

This extends a theorem of H. Heilbronn [4] and also sharpens a theorem of H. Davenport [3].

In this paper, which is a continuation of [4], the necessary theoretical background is given to enable the calculation of the irreducible Brauer projective characters of a given finite group to be carried out. As an example, this calculation is done for the alternating group A (7) in §3. In a future paper the calculations for the Mathieu groups will be presented.

In the earlier article [7], I began the study of rational period functions for the modular group Γ(l) = SL(2, Z) (regarded as a group of linear fractional transformations) acting on the Riemann sphere. These are rational functions q(z) which occur in functional equations of the form

where k∈Z and F is a function meromorphic in the upper half-plane ℋ, restricted in growth at the parabolic cusp ∞. The growth restriction may be phrased in terms of the Fourier expansion of F(z) at ∞:

with some μ∈Z. If (1.1) and (1.2) hold, then we call F a modular integral of weight 2k and q(z) the period of F.

Let H be a subgroup of Γ, the modular group. Let h be the number of orbits of under the action of H. In each orbit, the stabilizers are H-conjugate. Let U be the mapping z↦z + 1. Each stabilizer is Γ-conjugate to 〈Uc〉 for some non-negative integer c. The integer c is the cusp-width of the orbit. Let h0 be the number of orbits with non-trivial stabilizer, i.e. with c>0. The sequence (c(1), …, c(h0)) of non-zero cuspwidths is the cusp-split of H. Clearly, h0<h, and h∞ = h−h0 is the number of orbits with trivial stabilizer.

When Ramanujan died in 1920 he left behind three notebooks containing statements of a few thousand theorems, mostly without proofs. The second notebook is an enlarged edition of the first, and the third is short and fragmentary. Thus our primary attention may be directed toward the second notebook. In the decade following Ramanujan's death, G. N. Watson and B. M. Wilson agreed to perform the enormous task of editing the notebooks. Unfortunately, this task was never completed, possibly, in part, due to the premature death of Wilson in 1935. In 1957, a photostat edition [19] of the notebooks was published, but no editing whatsoever was undertaken.

In this note it is shown that if S is a free inverse semigroup of rank at least two and if e, f are idempotents of S such that e > f then S can be embedded in the partial semigroup eSe/fSf. The proof makes use of Scheiblich's construction for free inverse semigroups [7, 8] and of Reilly's characterisation of a set of free generators in an inverse semigroup [4, 5].