Related papers: Interpolation sets and nilsequences
We prove that if nonlinear complex polynomials of the same degree have orbits with infinite intersection, then the polynomials have a common iterate. We also prove a special case of a conjectured dynamical analogue of the Mordell-Lang…
The study of pinnacle sets has been a recent area of interest in combinatorics. Given a permutation, its pinnacle set is the set of all values larger than the values on either side of it. Largely inspired by conjectures posed by Davis,…
Pseudo-cones are a class of unbounded closed convex sets, not containing the origin. They admit a kind of polarity, called copolarity. With this, they can be considered as a counterpart to convex bodies containing the origin in the…
We evaluate the values of the Lebesgue constants in polynomial interpolation for three types of Cantor sets. In all cases, the sequences of Lebesgue constants are not bounded. This disproves the statement by Mergelyan.
In this paper, spindle starshaped sets are introduced and investigated, which apart from normalization form an everywhere dense subfamily within the family of starshaped sets. We focus on proving spindle starshaped analogues of recent…
An equivalent definition of the Fibonacci numbers is that they are the unique sequence such that every integer can be written uniquely as a sum of non-adjacent terms. We can view this as we have bins of length 1, we can take at most one…
A sequence $S=s_{1}s_{2}..._{n}$ is \emph{nonrepetitive} if no two adjacent blocks of $S$ are identical. In 1906 Thue proved that there exist arbitrarily long nonrepetitive sequences over 3-element set of symbols. We study a generalization…
This paper proves the minimum size of a supersequence over a set of eight elements is 52. This disproves a conjecture that the lower bound of the supersequence is the partial sum of the geometric Connell sequence. By studying the internal…
Although it is important both in theory as well as in applications, a theory of Birkhoff interpolation with main emphasis on the shape of the set of nodes is still missing. Although we will consider various shapes (e.g. we find all the…
Sumsets are central objects in additive combinatorics. In 2007, Granville asked whether one can efficiently recognize whether a given set $S$ is a sumset, i.e. whether there is a set $A$ such that $A+A=S$. Granville suggested an algorithm…
In this article we aim to develop from first principles a theory of sum sets and partial sum sets, which are defined analogously to difference sets and partial difference sets. We obtain non-existence results and characterisations. In…
We study families of possibly overlapping self-affine sets. Our main example is a family that can be considered the self-affine version of Bernoulli convolutions and was studied, in the non-overlapping case, by F.Przytycki and M.Urbanski.…
We prove that if a Bessel sequence in a Hilbert space, that is indexed by a countably infinite group in an invariant manner, can be partitioned into finitely many Riesz basic sequences, then each of the sets in the partition can be chosen…
In this note, we study a class of random subsets of positive integers induced by Bernoulli random variables. We obtain sufficient conditions such that the random set is almost surely lacunary, does not have bounded gaps and contains…
A Sidon set is a set of the positive integers such that the sums of two pairs is not repeated. I. Ruzsa gave a probabilistic construction of an infinite Sidon set. In this work we present the details of a simplified proof of this…
The concept of coreflexive set is introduced to study the structure of digraphs. New characterizations of line digraphs and nth-order line digraphs are given. Coreflexive sets also lead to another natural way of forming an intersection…
An optimal extension of the Jensen covering lemma, within the limits imposed by Prikry forcing, is proved. If L[E] is an "iterable" weasel with no measurable cardinals, then either L[E] has "indiscernibles", or every uncountable set of…
One of the most popular and studied recursive series is the Fibonacci sequence. It is challenging to see how Fibonacci numbers can be used to generate other recursive sequences. In our article, we describe some families of integer…
We study a strengthening of the notion of a perfectly meager set. We say that that a subset $A$ of a perfect Polish space $X$ is countably perfectly meager in $X$, if for every sequence of perfect subsets $\{P_n: n \in {\mathbb N}\}$ of…
In this paper, we prove some new thickness theorems with partial derivatives. We give some applications. First, we give a simple criterion that can judge whether two scaled Cantor sets have non-empty intersection. Second, we prove under…