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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 ∈ 1 \ 00 , one can consider the achievement set E( ) of all subsums of series ∞ =1 ( ). It is known that E( ) has one of the following structures: a finite union of closed intervals, a set homeomorphic to the Cantor set, a set homeomorphic to the set T of subsums of

We present a useful technique of proving strong c-algebrability. As an outcome we obtain the strong c-algebrability of the following sets of functions: strong Sierpiński-Zygmund, nowhere Hölder, Bruckner-Garg, nowhere monotone differentiable, a certain Baire class, smooth and nowhere analytic functions.1 2 ARTUR BARTOSZEWICZ, MAREK BIENIAS, MA LGORZATA FILIPCZAK, AND SZYMON G LA B have P (y 1 , ..., y k ) = 0 and P (y 1 , ..., y k ) ∈ A. This is a very natural way of proving algebrability, and therefore many authors, while proving algebrability, get strong algebrability actually.It is easy to check that for any cardinal number κ, the following implications hold.κ-strong algebrability ⇒ κ-algebrability ⇒ κ-lineability.Moreover, since every infinite dimensional Banach space has a linear base of cardinality c, then spaceability ⇒ c-lineability.On the other hand, none of these implications can be reversed (cf. [25,6]). Research on lineability, spaceability and algebrability can be viewed as a research on smallness of sets. The notions of smallness, like a measure zero, meager and σ-porosity, are connected with measure, topology and metric structure, respectively. The set that is not lineable, spaceable or algebrable can be viewed as small, since it does not contain a large algebraic structure. It means that being nonlineable, nonspaceable or nonalgebrable is an algebraic notion of smallness. The above named notions of smallness are not only different from the algebraic ones, but they are also incomparable. The following example shows, that largeness in algebrability does not coincide with largeness in topology. In [27] it was proved that the set of all functions from C[0, 1] which attain their maximum at exactly one point and attain their minimum at exactly one point is not 2-lineable. On the other hand, this set is residual. In [8] it was shown that there is a subset of ℓ 1 which is residual, strongly c-algebrable, but not spaceable. It is well known that every proper closed infinitely dimensional subspace of a Banach space is porous, which shows that a spaceable set can be very small from topological and metric point of view.In this paper we prove several results in strong c-algebrability. Almost all the presented below results are the best possible in the sense of complexity of the structure and cardinality of the set of generators. Let us remark, that the he c-algebrability of functions families F, with |F| = c, that is maximal algebrability, was also proved in a few papers - [25], [26] and [24].The paper is organized as follows. In Section 2 we prove that family of all strongly Sierpiński-Zygmund functions is strongly c-algebrable provided this family is nonempty. Strongly Sierpiński-Zygmund functions are classical Sierpiński-Zygmund functions under the Continuum Hypothesis, they do not exist under Martin's Axiom (MA + ¬CH), and there are strongly Sierpiński-Zygmund functions in some models of ZFC in which the Continuum Hypothesis fails. In Section 3 we prove strong c-algebrability of family of all functions f :...