The paper deals with the sets of numbers from [0,1] such that their binary representation is almost convergent. The aim of the study is to compute the Hausdorff dimensions of such sets. Previously, the results of this type were proved for a single summation method (e.g. Cesàro, Abel, Toeplitz). This study extends the results to a wide range of matrix summation methods.

Recently J. B. Reade determined the spectrum of C, the Cesaro matrix of order 1, considered as an operator on cO, the space of null sequences. Previously F. P. A. Cass and the author had determined the spectra for a large class of weighted mean operators on c, the space of convergent sequences. Subsequently the author determined the fine spectra of these operators over c. This paper examines the spectra and fine spectra of weighted mean operators on cO, obtaining the result of Reade as a special case.

The (F, dn) method is investigated with respect to perfectness, strong regularity, and summing bounded divergent sequences. In the process the columns of the inverse matrix are characterized in terms of {dn}.

Necessary conditions and sufficient conditions are obtained for the generalized Lototsky summability matrix method (F, dn) to be regular and Mercerian. In particular, a set of conditions equivalent to being regular and Mercerian is given for real {dn} and for complex {dn} eventually in any closed half-plane containing the origin.

Some inclusion theorems are obtained relating the absolute summability of divergent integrals of the form under three summability methods: Abelian A(x), Abelian A(lnx) and Stieltjes S(x).

In this paper an explicit regular sequence-to-sequence summability method is presented which sums the geometric series to the value 1/(1-z) in all of ℂ\{1} and to infinity at the point 1. The method also provides compact convergence in ℂ \ [ 1, ∞) and therefore improves well-known results by Le Roy, Lindelöf and Mittag-Leffler.