We introduce a notion of strong algebrability of subsets of linear algebras. Our main results are the following. The set of all sequences from c 0 which are not summable with any power is densely strongly c-algebrable. The set of all sequences in l ∞ whose sets of limit points are homeomorphic to the Cantor set is comeager and strongly c-algebrable. The set of all nonmeasurable functions from R R is 2 c-algebrable. These results complete several by other authors, within the modern context of lineability.

Abstract. Given a finite subset Σ ⊂ R and a positive real number q < 1 we study topological and measure-theoretic properties of the self-similaranq n : (an)n∈ω ∈ Σ ω , which is the unique compact solution of the equation K = Σ + qK. The obtained results are applied to studying partial sumsets E(x) = ∞ n=0 xnεn : (εn)n∈ω ∈ {0, 1} ω of multigeometric sequences x = (xn)n∈ω. Such sets were investigated by Ferens, Morán, Jones and others. The aim of the paper is to unify and deepen existing piecemeal results.

Let CBV denote the Banach algebra of all continuous real-valued functions of bounded variation, defined in [0,1]. We show that the set of strongly singular functions in CBV is nonseparably spaceable. We also prove that certain families of singular functions constitute strongly c-algebrable sets. The argument is based on a new general criterion of strong c-algebrability. IntroductionIn the last decade, much work was done in the study of subsets of vector spaces (topological vector spaces, normed spaces, Banach algebras, etc.) with no linear structure given a priori. This research was earlier initiated by Gurariy [13], [14] and then continued by several authors.Recall (see [1]) that, for a topological vector space V , its subset A is said to be:• spaceable if A ∪ {0} contains an infinite dimensional closed vector subspace W of V ; moreover, if W is nonseparable, we say that A is nonseparably spaceable.One aim of our paper is to reexamine the spaceability of some families of singular functions contained in the Banach algebra CBV of all continuous functions from [0, 1] to R of bounded variation, endowed with the norm ||f || = |f (0)| + Var (f ) where Var(f ) = Var [0,1] f denotes the total variation of f in [0, 1] and, in general, Var [a,b] f denotes the variation of f in a subinterval [a, b] of [0, 1]. The lineability and spaceability of certain subfamilies of CBV were studied in [5] and more recently, in [6]. Our main result going in this direction states that the set of strongly singular functions is nonseparably spaceable (Theorem 2). We heavily exploit a family of such functions known in the probability theory [8].Another aim of our paper is to establish strong c-algebrability of some families of singular functions. Here c denotes the cardinality of R (continuum). Algebrability and strong algebrability are associated with algebras, the structures richer than linear spaces. The notion of strong algebrability for various special subfamilies of CBV, C[0, 1] and R [0,1] becomes interesting in 1991 Mathematics Subject Classification. 15A03, 26A30, 26A45, 46B25, 46E25, 46J15.

For a sequence x ∈ ℓ1 \ c00, one can consider the set E(x) of all subsums of series ∞ n=1 x(n). Guthrie and Nymann proved that E(x) is one of the following types of sets:(I) a finite union of closed intervals;(C) homeomorphic to the Cantor set;(MC) homeomorphic to the set T of subsums of ∞ n=1 b(n) where b(2n − 1) = 3/4 n and b(2n) = 2/4 n .By I, C and MC denote the sets of all sequences x ∈ ℓ1 \ c00, such that E(x) has the property (I), (C) and (MC), respectively. In this note we show that I and C are strongly c-algebrable and MC is c-lineable. We show that C is a dense G δ -set in ℓ1 and I is a true Fσ-set. Finally we show that I is spaceable while C is not spaceable.