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We show, from a topological viewpoint, that most numbers are not normal in a strong sense. More precisely, the set of numbers $x \in (0,1]$ with the following property is comeager: for all integers $b\ge 2$ and $k\ge 1$ , the sequence of vectors made by the frequencies of all possibile strings of length k in the b-adic representation of x has a maximal subset of accumulation points, and each of them is the limit of a subsequence with an index set of nonzero asymptotic density. This extends and provides a streamlined proof of the main result given by Olsen (2004) in this Journal. We provide analogues in the context of analytic P-ideals and regular matrices.

No abstract

We show, from a topological viewpoint, that most numbers are not normal in a strong sense. More precisely, the set of numbers x ∈ (0, 1] with the following property is comeager: for all integers b ≥ 2 and k ≥ 1, the sequence of vectors made by the frequencies of all possibile strings of length k in the b-adic representation of x has a maximal subset of accumulation points, and each of them is the limit of a subsequence with an index set of nonzero asymptotic density. This extends and provides a streamlined proof of the main result given by Olsen in [Math. Proc. Cambridge Philos. Soc. 137 (2004), 43-53]. We provide analogues in the context of analytic P-ideals and regular matrices.

We present three classes of abstract prearithmetics, tA M u Mě1 , tA ´M,M u Mě1 , and tB M u Mě0 . The first one is weakly projective with respect to the conventional nonnegative real Diophantine arithmetic R `" pR `, `, ˆ, ď R`q , while the other two are weakly projective with respect to the conventional real Diophantine arithmetic R " pR, `, ˆ, ď R q.In addition, we have that every A M and every B M are a complete totally ordered semiring, while every A ´M,M is not. We show that the weak projection of any series in R `converges in A M , for any M ě 1, and that the weak projection of any non-oscillating series in R converges in A ´M,M , for any M ě 1, and in B M , for all M P R `. We also prove that working in A M and in A ´M,M , for any M ě 1, allows to overcome a version of the paradox of the heap, while working in B M does not. Introduction.Although the conventional arithmetic-which we call Diophantine from Diophantus, the Greek mathematician who first approached this branch of mathematics-is almost as old as mathematics itself, it sometimes fails to correctly describe natural phenomena. For example, in [3] Helmoltz points out that adding one raindrop to another one leaves us with one raindrop, while in [10] Kline notices that Diophantine arithmetic fails to correctly describe the result of combining gases or liquids by volume. Indeed, one quarter of alcohol and one quarter of water only yield about 1.8 quarters of vodka. To overcome this issue, scholars started developing inconsistent arithmetics, that is, arithmetics for which one or more Peano axioms were at the same time true and false. The most striking one was ultraintuitionism, developed by Yesenin-Volpin in [11], that asserted that only a finite quantity of natural numbers exists. Other authors suggested that numbers are finite (see e.g. [2] and [4]), while different scholars adopted a more moderate approach. The inconsistency of these alternative arithmetics lies in the fact that they are all grounded in the ordinary Diophantine arithmetic. The first consistent alternative to Diophantine arithmetic was proposed by Burgin [6], and the name non-Diophantine seemed perfectly suited for this arithmetic. Non-Diophantine arithmetics for natural and whole numbers have been studied by Burgin in [6,7,8,9], while those for real and complex numbers by Czachor in [1,5].There are two types of non-Diophantine arithmetics: dual and projective. In this paper, we work with the latter. We start by defining an abstract prearithmetic A,