HAMEL SPACES AND DISTAL EXPANSIONS

In this note, we construct a distal expansion for the structure $$\left( {; + , < ,H} \right)$$, where $H \subseteq $ is a dense $Q$-vector space basis of $R$ (a so-called Hamel basis). Our construction is also an expansion of the dense pair $\left( {; + , < ,} \right)$ and has full quantifier elimination in a natural language.

(B∆I,I)-saturated sets and Hamel basis

Let I be a proper σ-ideal of subsets of the real line. In a σ-field of Borel sets modulo sets from the σ-ideal I we introduce an analogue of the saturated non-measurability considered by Halperin. Properties of (B∆I,I)-saturated sets are investigated. M. Kuczma considered a problem how small or large a Hamel basis can be. We try to study this problem in the context of sets from I.

On some Problem of Sierpiński and Ruziewicz Concerning the Superposition of Measurable Functions. Microscopic Hamel Basis

S. Ruziewicz and W. Sierpiński proved that each function f : ℝ → ℝ can be represented as a superposition of two measurable functions. Here, a strengthening of this theorem is given. The properties of Lusin set and microscopic Hamel bases are considered

On some properties of Hamel bases and their applications to Marczewski measurable functions

We introduce new properties of Hamel bases. We show that it is consistent with ZFC that such Hamel bases exist. Under the assumption that there exists a Hamel basis with one of these properties we construct a discontinuous and additive function that is Marczewski measurable. Moreover, we show that such a function can additionally have the intermediate value property (and even be an extendable function). Finally, we examine sums and limits of such functions.

On Functions Whose Graph is a Hamel Basis, II

We say that a functionh: ℝ → ℝ is a Hamel function (h∈ HF) ifh, considered as a subset of ℝ2, is a Hamel basis for ℝ2. We show that A(HF) ≥ ω,i.e.,for every finiteF⊆ ℝℝthere existsf∈ ℝℝsuch thatf+F⊆ HF. From the previous work of the author it then follows that A(HF) = ω.