相关论文: Sets with distinct sums of pairs, long arithmetic …

It is known that if a subset of $\mathbb{R}$ has positive Lebesgue measure, then it contains arbitrarily long finite arithmetic progressions. We prove that this result does not extend to infinite arithmetic progressions in the following…

Here we consider piecewise fractional linear maps with three branches. The paper presents a study of invariant measures with densities which can be written as infinite series. These series either have infinitely many poles or they sum up to…

If $a$ and $b$ are integers with $b>a>1$, we completely characterize ``long'' arithmetic progressions in the sumsets of the geometric progressions $1, a, a^2, a^3, \ldots$ and $1, b, b^2, b^3, \ldots$. Our proofs utilize recent applications…

L^p spaces of mappings taking values in arbitrary metric spaces, which we call nonlinear Lebesgue spaces, play an important role in several fields of mathematics. For instance, membership in these spaces is typically required for transport…

Let $\varphi(x_1,\ldots,x_h,y) = u_1x_1 + \cdots + u_hx_h+vy$ be a linear form with nonzero integer coefficients $u_1,\ldots, u_h, v.$ Let $\mathcal{A} = (A_1,\ldots, A_h)$ be an $h$-tuple of finite sets of integers and let $B$ be an…

In [8] we found a class of overlapping asymmetric self-similar measures on the real line, which are generically absolutely continuous with respect to the Lebesgue measure. Here we construct exceptional measures in this class being singular.

We consider nonlinear, or "event-dependent", sampling, i.e. such that the sampling instances {tk} depend on the function being sampled. The use of such sampling in the construction of Lebesgue's integral sums is noted and discussed as…

Suppose $a_n$ is a real, nonnegative sequence that does not increase exponentially. For any $p<1$ we contruct a Lebesgue measurable set $E \subseteq \mathbb{R}$ which has measure at least $p$ in any unit interval and which contains no…

An alternative mathematics based on qualitative plurality of finiteness is developed to make non-standard mathematics independent of infinite set theory. The vague concept "accessibility" is used coherently within finite set theory whose…

For continuous maps on a compact manifold M, particularly for those that do not preserve the Lebesgue measure m, we define the observable invariant probability measures as a generalization of the physical measures. We prove that any…

It is proved the existence of large algebraic structures \break --including large vector subspaces or infinitely generated free algebras-- inside, among others, the family of Lebesgue measurable functions that are surjective in a strong…

For a certain parametrized family of maps on the circle, with critical points and logarithmic singularities where derivatives blow up to infinity, a positive measure set of parameters was constructed in [19], corresponding to maps which…

Classical theorems from the early 20th century state that any Haar measurable homomorphism between locally compact groups is continuous. In particular, any Lebesgue-measurable homomorphism $\phi:\mathbb{R} \to \mathbb{R}$ is of the form…

We analyze certain parametrized families of one-dimensional maps with infinitely many critical points from the measure-theoretical point of view. We prove that such families have absolutely continuous invariant probability measures for a…

We consider a family of Pomeau-Manneville type interval maps $T_\alpha$, parametrized by $\alpha \in (0,1)$, with the unique absolutely continuous invariant probability measures $\nu_\alpha$, and rate of correlations decay $n^{1-1/\alpha}$.…

We show that for every $\varepsilon>0$ there is an absolute constant $c(\varepsilon)>0$ such that the following is true. The union of any $n$ arithmetic progressions, each of length $n$, with pairwise distinct differences must consist of at…

We analyze a class of piecewise linear parabolic maps on the torus, namely those obtained by considering a linear map with double eigenvalue one and taking modulo one in each component. We show that within this two parameter family of maps,…

I provide a proof of the existence of absolutely continuous invariant measures (and study their statistical properties) for multidimensional piecewise expanding systems with not necessarily bounded derivative or distortion. The proof uses…

Consider the self-map F of the space of real-valued test functions on the line which takes a test function f to the test function sending a real number x to f(f(x))-f(0). We show that F is discontinuous, although its restriction to the…

Several Lebesgue-type decomposition theorems in analysis have a strong relation to the operation called: parallel sum. The aim of this paper is to investigate this relation from a new point of view. Namely, using a natural generalization of…