The aim of this work is to prove new results on a class of digital functions with special emphasis on shifted primes as arguments. Our method lies on the estimate of exponential sums of the form $\sum_{n\leq x}\Lambda (n)\exp(2i\pi f(n+c_n)+\beta n)$ where $f$ a digital function, $\mathbf{c}=(c_n)$ is an almost-periodic sequence in $ \mathbb{Z}$ and $\beta $ is a real parameter, which extend the works of Mauduit-Rivat \cite{mr1} and Martin-Mauduit-Rivat \cite{mmr} to the case of the shifted prime numbers satisfying a digital constraint.

It is shown that for the vector space $\mathbb{R^N}$ (equipped with the product topology and the Yamasaki-Kharazishvili measure) the group of linear measure preserving isomorphisms is quite rich. Using Kharazishvili's approach, we prove that every infinite-dimensional Polish linear space admits a $\sigma$-finite non-trivial Borel measure that is translation invariant with respect to a dense linear subspace. This extends a~recent result of Gill, Pantsulaia and Zachary on the existence of such measures in Banach spaces with Schauder bases. It is shown that each $\sigma$-finite Borel measure defined on an infinite-dimensional Polish linear space, which assigns the value 1 to a fixed compact set and is translation invariant with respect to a~linear subspace fails the uniqueness property. For Banach spaces with absolutely convergent Markushevich bases, a similar problem for the usual completion of the concrete $\sigma$-finite Borel measure is solved positively. The uniqueness problem for non-$\sigma$-finite semi-finite translation invariant Borel measures on a Banach space $X$ which assign the value 1 to the standard rectangle (i.e., the rectangle generated by an absolutely convergent Markushevich basis) is solved negatively. In addition, it is constructed an example of such a measure $\mu_B^0$ on $X$, which possesses a strict uniqueness property in the class of all translation invariant measures which are defined on the domain of $\mu_B^0$ and whose values on non-degenerate rectangles coincide with their volumes.

The criteria for non-containment of $l^1$ for the classes of Banach and Fréchet spaces are extended to the class of locally complete locally convex spaces the bounded sets of which are metrizable.

Let $X$ be a real normed vector space and $\mathcal{B}$$(X)$ be the cone of all nonempty bounded closed convex subsets of $X$. For $A,B,C,D\in \mathcal{B}$$(X)$ we have a relation of equivalence defined by $(A,B)\sim (C,D)$ if and only if $\overline{A+D}=\overline{B+C}$. By $[A,B]$ we denote the quotient class of $(A,B)$. The quotient space $\widetilde{X}=\mathcal{B}$$^{2}(X)/_{\sim} $ is a vector space called the Minkowski-Rådström-Hörmander space over $X$. For $\tilde{x}=[A,B]\in \widetilde{X}$ we have the Hausdorff norm $\|\tilde{x}\|_{H}=d_{H}(A,B)=\inf \{\varepsilon >0|\, A\subset B+\varepsilon \mathbb{B},B\subset A+\varepsilon \mathbb{B}\}$ where $\mathbb{B}$ is the closed unit ball in $X$. We also define Bartels-Pallaschke norm $\|\tilde{x}\|_{BP}=\inf \{\|C\|+\|D\|\,|\,(C,D)\in [A,B] \}$, where $\|A\|=\sup_{a\in A}\|a\|$. In this paper we prove that the bilinear function $(\cdot,\cdot):(\widetilde{\mathbb{R}^{2}},\|\cdot\|_{H})\times (\widetilde{\mathbb{R}^{2}},\|\cdot\|_{BP}) \longrightarrow \mathbb{R}$ defined by $(\tilde{x},\tilde{y})=2V(\tilde{x},\tilde{y})+\langle s\tilde{x},s\tilde{y}\rangle$, where $V(\tilde{x},\tilde{y})$ is a generalized mixed volume and $s\tilde{x}$ is a generalized Steiner's point, satisfies the inequality $|(\tilde{x},\tilde{y})|\leq (2\pi +1)\|\tilde{x}\|_{H}\|\tilde{y}\|_{BP}$. We also prove that this bilinear function defines an isomorphic mapping between Banach spaces $(\widetilde{\mathbb{R}^{2}},\|\cdot \|_{BP})$ and the dual space to $(\widetilde{\mathbb{R}^{2}},\|\cdot \|_H)$ (Theorem 2).

We deal with decomposition theorems for measures on Boolean algebras or -- more generally -- for modular functions on lattices. In particular, for measures on Boolean algebras with values in locally solid $\ell$-groups we compare decomposition theorems obtained with the Frechét-Nikodým-approach and decomposition theorems obtained using the band decomposition theorem of Riesz.

It is difficult to write a paper on a mathematical work of a person like Lech Drewnowski.First of all because of a large variety of subjects he worked on.Secondly, since several of his topics which seem to be completely pairwisely unrelated surprisingly turn out to be connected on a deep level: look for instance, on Drewnowski's work on Nikodým and Vitali-Hahn-Saks theorems -it is a root of his interest in barrelled spaces and one can easily observe its further interrelation with his work on series convergence in the context of the Orlicz-Pettis theorem.The third reason why our task is close to impossible is that Drewnowski's work is far from being completed: every year new and thorough papers appear so one cannot predict which of that existing papers should be reinterpreted in a new light of his recent discoveries.Last but not least it is difficult to write about our own Master and Teacher whom we owe so much.We do not present a systematic and detailed analysis of all Drewnowski's results.We have subjectively selected several topics and theorems due to him which are, in our private opinion, beautiful, important and also sufficiently near to our own interests.Let us say that Drewnowski is a true "problem solver".When others feel that the solution is already complete and fully satisfactory, he works further and finds unexpected deeper results and he builds an extensive theory.While collaborating we experienced this phenomena several times.One could say that the "first solution" is only a motivation and starting point for Drewnowski's extensive research which leads to the "deep solution".In this paper we explain a little bit of motivation of the particular theorems but the emphasis will be put on further results inspired by Drewnowski's research.We will show how many people owe their inspiration to Drewnowski.Writing this survey we realized once again how lovely Lech's mathematics is.

We are concerned with the notions of absolute continuity in terms of $0$-continuity, and of $(\epsilon,\delta)$-continuity, with respect to a finitely additive function taking values into a topological commutative group. A sufficient condition for their equivalence is established. The non-additive case of functions with values into Hausdorff topological spaces is also investigated.

In this paper, we study the structure of $End(M)$ where $M$ is a motive of $GL_2$-type over a totally real field. Under some standard conjectures for motives, we then prove some results about Galois representations attached to such motives.

In this paper we study the four-dimensional divisor problem of $(a,b,c,c)$ type, where $1\leq a\leq b<c$ are fixed integers. Our theorems improve some classical results. As an application, we study the error term of the summatory function of the exponential totient function.

For k 2 we bound the number of pairs of consecutive k-full numbers.