Sets Of Positive Upper Density Research Articles

On the metric upper density of Birkhoff sums for irrational rotations

This article examines the value distribution of for almost every α where is ranging over a long interval and f is a 1-periodic function with discontinuities or logarithmic singularities at rational numbers. We show that for N in a set of positive upper density, the order of is of Khintchine-type, unless the logarithmic singularity is symmetric. Additionally, we show the asymptotic sharpness of the Denjoy–Koksma inequality for such f, with applications in the theory of numerical integration. Our method also leads to a generalized form of the classical Borel–Bernstein Theorem that allows very general modularity conditions.

D-finiteness, rationality, and height II: Lower bounds over a set of positive density

We consider D-finite power series f(z)=∑n≥0anzn with coefficients in a number field K. We show that there is a dichotomy governing the behaviour of h(an) as a function of n, where h is the absolute logarithmic Weil height. As an immediate consequence of our results, we have that either f(z) is rational or h(an)>[K:Q]−1⋅log⁡(n)+O(1) for n in a set of positive upper density and this is best possible when K=Q.

Boxes, extended boxes and sets of positive upper density in the Euclidean space

AbstractWe prove that sets with positive upper Banach density in sufficiently large dimensions contain congruent copies of all sufficiently large dilates of three specific higher-dimensional patterns. These patterns are: 2n vertices of a fixed n-dimensional rectangular box, the same vertices extended with n points completing three-term arithmetic progressions, and the same vertices extended with n points completing three-point corners. Our results provide common generalizations of several Euclidean density theorems from the literature.

Polynomial configurations in sets of positive upper density over local fields

Let F(x) = (f1(x), …, fm(x)) be such that 1, f1, …, fm are linearly independent polynomials with real coefficients. Based on ideas of Bachoc, DeCorte, de Oliveira and Vallentin in combination with estimating certain oscillatory integrals with polynomial phase we will show that the independence ratio of the Cayley graph of ℝm with respect to the portion of the graph of F defined by a ≤ log åså ≤ T is at most O(1/(T — a)). We conclude that if I ⊆ ℝm has positive upper density, then the difference set I — I contains vectors of the form F(s)for an unbounded set of values s ∈ ℝ. It follows that the Borel chromatic number of the Cayley graph of ℝm with respect to the set {±F(s): s ∈ ℝ} is infinite. Analogous results are also proven when ℝ is replaced by the field of p-adic numbers ℚp. At the end, we will also show the existence of real analytic functions f1, …, fm, for which the analogous statements no longer hold.

Approximation by special values of Dirichlet series

In this note, we will show that real numbers can be strongly approximated by linear combinations of special values of Dirichlet series. We extend the approximation results of Emre Alkan in an effective way to all non-zero Dirichlet series with a better approximation. Using the fundamental works of Szemerédi and Green-Tao on arithmetic progressions, we prove that one can approximate real numbers with special values of Dirichlet series coming from sets of positive upper density or the set of prime numbers.

Which subsets of an infinite random graph look random?

AbstractGiven a countable graph, we say a set A of its vertices is universal if it contains every countable graph as an induced subgraph, and A is weakly universal if it contains every finite graph as an induced subgraph. We show that, for almost every graph on , (1) every set of positive upper density is universal, and (2) every set with divergent reciprocal sums is weakly universal. We show that the second result is sharp (i.e., a random graph on will almost surely contain non‐universal sets with divergent reciprocal sums) and, more generally, that neither of these two results holds for a large class of partition regular families.

Simplices and sets of positive upper density in $\mathbb {R}^d$

We prove an extension of Bourgain's theorem on pinned distances in measurable subset of $\mathbb{R}^2$ of positive upper density, namely Theorem $1^\prime$ in [Bourgain, 1986], to pinned non-degenerate $k$-dimensional simplices in measurable subset of $\mathbb{R}^{d}$ of positive upper density whenever $d\geq k+2$ and $k$ is any positive integer.

Multiple recurrence and convergence along the primes

Let $E\subset \Bbb{Z}$ be a set of positive upper density. Suppose that $P_1,P_2,\ldots,P_k\in {\Bbb Z}[X]$ are polynomials having zero constant terms. We show that the set $E\cap (E-P_1(p-1))\cap\cdots\cap (E-P_k(p-1))$ is non-empty for some prime number $p$. Furthermore, we prove convergence in $L^2$ of polynomial multiple averages along the primes.

Subsets of products of finite sets of positive upper density

In this note we prove that for every sequence (mq)q of positive integers and for every real 0<δ≤1 there is a sequence (nq)q of positive integers such that for every sequence (Hq)q of finite sets such that |Hq|=nq for every q∈N and for every D⊆⋃k∏q=0k−1Hq with the property thatlimsupk|D∩∏q=0k−1Hq||∏q=0k−1Hq|≥δ there is a sequence (Jq)q, where Jq⊆Hq and |Jq|=mq for all q, such that ∏q=0k−1Jq⊆D for infinitely many k. This gives us a density version of a well-known Ramsey-theoretic result. We also give some estimates on the sequence (nq)q in terms of the sequence of (mq)q.

The Wildcard Set's Size in the Density Hales-Jewett Theorem

In this paper prove results concerning restrictions on the cardinality of the wildcard set in the density Hales-Jewett theorem, establishing in particular that for general $k$ one may choose this cardinality from any IP set and that for $k=2$ it may be chosen to be a square, thus providing an abstract extension of Sárközy's theorem on square differences in sets of positive upper density.

Nil–Bohr sets of integers

AbstractWe study relations between subsets of integers that are large, where large can be interpreted in terms of size (such as a set of positive upper density or a set with bounded gaps) or in terms of additive structure (such as a Bohr set). Bohr sets are fundamentally abelian in nature and are linked to Fourier analysis. Recently it has become apparent that a higher order, non-abelian, Fourier analysis plays a role both in additive combinatorics and in ergodic theory. Here we introduce a higher-order version of Bohr sets and give various properties of these objects, generalizing results of Bergelson, Furstenberg, and Weiss.

Sumset phenomenon in countable amenable groups

Jin proved that whenever A and B are sets of positive upper density in Z , A + B is piecewise syndetic. Jin's theorem was subsequently generalized by Jin and Keisler to a certain family of abelian groups, which in particular contains Z d . Answering a question of Jin and Keisler, we show that this result can be extended to countable amenable groups. Moreover we establish that such sumsets (or — depending on the notation — “product sets”) are piecewise Bohr, a result which for G = Z was proved by Bergelson, Furstenberg and Weiss. In the case of an abelian group G, we show that a set is piecewise Bohr if and only if it contains a sumset of two sets of positive upper Banach density.

From Bi-Immunity to Absolute Undecidability

AbstractAn infinite binary sequence A is absolutely undecidable if it is impossible to compute A on a set of positions of positive upper density. Absolute undecidability is a weakening of bi-immunity. Downey, Jockusch and Schupp [2] asked whether, unlike the case for bi-immunity, there is an absolutely undecidable set in every non-zero Turing degree. We provide a positive answer to this question by applying techniques from coding theory. We show how to use Walsh–Hadamard codes to build a truth-table functional which maps any sequence A to a sequence B, such that given any restriction of B to a set of positive upper density, one can recover A. This implies that if A is non-computable, then B is absolutely undecidable. Using a forcing construction, we show that this result cannot be strengthened in any significant fashion.

Nilfactors of ℝ m and configurations in sets of positive upper density in ℝ m

We use ergodic theoretic tools to solve a classical problem in geometric Ramsey theory. Let E be a measurable subset of R^m, with positive upper density. Let V={0,v_1,...,v_k} be a subset of R^m. We show that for r large enough, we can find an isometric copy of rV arbitrarily close to E. This is a generalization of a theorem of Furstenberg, Katznelson and Weiss showing a similar property for m=k=2.

Multiplicatively large sets and ergodic Ramsey theory

Multiplicatively large sets are defined in (ℕ, ·) by an analogy to sets of positive upper density in (ℕ, +). By utilizing various ergodic multiple recurrence theorems, we show that multiplicatively large sets have a rich combinatorial structure. In particular, it is proved that for any multiplicatively large setE ⊂ ℕ and anyk ∈ ℕ, there existsa,b,c,d,e,q ∈ ℕ such that {fx23-1}

Translation Properties of Sets of Positive Upper Density

Generalizing a result of Raimi we show that there exists a set $E \subset {\mathbf {N}}$ such that if $A \subset {\mathbf {N}}$ is a set with positive upper density, then there exists a number $k \in {\mathbf {N}}$ such that ${d^ * }((A + k) \cap E) > 0$ and ${d^ * }((A + k) \cap {E^c}) > 0$. Some extensions and further results are also obtained.

Translation properties of sets of positive upper density

Generalizing a result of Raimi we show that there exists a set E ⊂ N E \subset {\mathbf {N}} such that if A ⊂ N A \subset {\mathbf {N}} is a set with positive upper density, then there exists a number k ∈ N k \in {\mathbf {N}} such that d ∗ ( ( A + k ) ∩ E ) &gt; 0 {d^ * }((A + k) \cap E) &gt; 0 and d ∗ ( ( A + k ) ∩ E c ) &gt; 0 {d^ * }((A + k) \cap {E^c}) &gt; 0 . Some extensions and further results are also obtained.