Sets Of Positive Density Research Articles

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.

The Hasse principle for finite Galois modules allowing exceptional sets of positive density

We study a variant of the Hasse principle for finite Galois modules, allowing exceptional sets of positive density. For a Galois module whose underlying abelian group is isomorphic to [Formula: see text] ([Formula: see text]), we show that the product of the restriction maps for places in a set of places [Formula: see text] is injective if the Dirichlet density of [Formula: see text] is strictly larger than [Formula: see text]. We give applications to the local–global divisibility problem for elliptic curves and the Hasse principle for flexes on plane cubic curves.

Patterns in sets of positive density in trees and affine buildings

We prove an analogue for homogeneous trees and certain affine buildings of a result of Bourgain on pinned distances in sets of positive density in Euclidean spaces. Furthermore, we construct an example of a non-homogeneous tree with positive Hausdorff dimension, and a subset with positive density thereof, in which not all sufficiently large (even) distances are realised.

On the limits of real-valued functions in sets involving ψ-density, and applications

New results on upper and lower limits of real-valued functions are proved by means of ψ-densities introduced by P. D. Barry in 1962. This leads to improvements of several existing results on the growth of non-decreasing and unbounded real-valued functions in sets of positive density. The ψ-densities allow us to introduce a new concept of a limit for real-valued functions, which is used to reveal further properties on the behavior of integrable functions at infinity.

A note on configurations in sets of positive density which occur at all large scales

Furstenberg, Katznelson and Weiss proved in the early 1980s that every measurable subset of the plane with positive density at infinity has the property that all sufficiently large real numbers are realised as the Euclidean distance between points in that set. Their proof used ergodic theory to study translations on a space of Lipschitz functions corresponding to closed subsets of the plane, combined with a measure-theoretical argument. We consider an alternative dynamical approach in which the phase space is given by the set of measurable functions from ℝd to [0, 1], which we view as a compact subspace of L ∞(ℝd) in the weak-* topology. The pointwise ergodic theorem for ℝd-actions implies that with respect to any translation-invariant measure on this space, almost every function is asymptotically close to a constant function at large scales. This observation leads to a general sufficient condition for a configuration to occur in every set of positive upper Banach density at all sufficiently large scales. To illustrate the use of this criterion we apply it to prove a new result concerning three-point configurations in measurable subsets of the plane which form the vertices of a triangle with specified area and side length, yielding a new proof of a result related to work of R. Graham.

Nonconventional ergodic averages and multiple recurrence for von Neumann dynamical systems

The Furstenberg recurrence theorem (or equivalently, Szemer\'edi's theorem) can be formulated in the language of von Neumann algebras as follows: given an integer $k \geq 2$, an abelian finite von Neumann algebra $(\M,\tau)$ with an automorphism $\alpha: \M \to \M$, and a non-negative $a \in \M$ with $\tau(a)>0$, one has $\liminf_{N \to \infty} \frac{1}{N} \sum_{n=1}^N \Re \tau(a \alpha^n (a) ... \alpha^{(k-1)n} (a)) > 0$; a subsequent result of Host and Kra shows that this limit exists. In particular, $\Re \tau(a \alpha^n (a) >... \alpha^{(k-1)n} (a)) > 0$ for all $n$ in a set of positive density. From the von Neumann algebra perspective, it is thus natural to ask to what extent these results remain true when the abelian hypothesis is dropped. All three claims hold for $k = 2$, and we show in this paper that all three claims hold for all $k$ when the von Neumann algebra is asymptotically abelian, and that the last two claims hold for $k=3$ when the von Neumann algebra is ergodic. However, we show that the first claim can fail for $k=3$ even with ergodicity, the second claim can fail for $k \geq 4$ even assuming ergodicity, and the third claim can fail for $k=3$ without ergodicity, or $k \geq 5$ and odd assuming ergodicity. The second claim remains open for non-ergodic systems with $k=3$, and the third claim remains open for ergodic systems with $k=4$.

A Furstenberg–Katznelson–Weiss type theorem on [formula omitted]-point configurations in sets of positive density in finite field geometries

We show that if E ⊂ F q d , the d -dimensional vector space over the finite field with q elements, and | E | ≥ ρ q d , where q − 1 2 ≪ ρ ≤ 1 , then E contains an isometric copy of at least c ρ d − 1 q d + 1 2 distinct ( d + 1 ) -point configurations.

Split reductions of simple abelian varieties

Consider an absolutely simple abelian variety X over a number field K. If the absolute endomorphism ring of X is commutative and satisfies certain parity conditions, then the reduction X_p is absolutely simple for almost all p. Conversely, if the absolute endomorphism ring of X is noncommutative, then X_p is reducible for p in a set of positive density.

On the zeros of recurrence sequences with non-constant coefficients

The paper deals with the zeros of sequences, which satisfy linear recurrences with non-constant coefficients. We prove a weaker analogon of the Skolem-Mahler-Lech theorem on ordinary recurrent sequences. The proof relies on Szemeredi's theorem on arithmetic progressions in sets of positive density.

On a Theorem of Plünnecke Concerning the Sum of A Basis and A Set of Positive Density

In 1935 Erdős proved that every additive basis B of order h is an essential component by showing σ( A + B ) ≥ σ( A ) + (1/2 h) σ( A )(1 − σ( A )), where σ denotes Schnirelmann density. This lower bound was improved by Plünnecke to σ( A + B ) ≥ σ( A ) 1 − 1/ h , giving the best exponent. A simplified proof is presented.

A szemerédi type theorem for sets of positive density inR k

Letk≧2 andA a subset ofRk of positive upper density. LetV be the set of vertices of a (non-degenerate) (k−1)-dimensional simplex. It is shown that there existsl=l(A, V) such thatA contains an isometric image ofl′. V wheneverl′>l. The casek=2 yields a new proof of a result of Katznelson and Weiss [4]. Using related ideas, a proof is given of Roth’s theorem on the existence of arithmetic progressions of length 3 in sets of positive density.

On the number of groups of a given order

Letting G( n) denote the number of nonisomorphic groups of order n, it is shown that for square-free n, G( n) ≤ ϕ( n) and G( n) ≤ (log n) c on a set of positive density. Letting F k ( x) denote the number of n ≤ x for which G( n) = k, it is shown that F 2(x) = O(x( log 4x) ( log 3x) 2) , where log r x denotes the r-fold iterated logarithm.

A generalization of jarnik's theorem on diophantine approximations

Let s be a positive integer, 0⩽ v⩽1, L any subset of positive integers such that ∑ qϵl q −v−ε is divergent but ∑ qϵl q −v−ε is convergent for every ε>0. Let λ>1+ν/ s and denote by E λ (L) the set of all real s-tuples (α 1,…,α s ) satisfying the set of inequalities | qx i |≤ q 1− λ ( i=1,…, s) for an infinite number of qϵL. (|α|) denotes the distance from x to the nearest integer.) It is proved that the Hausdorff dimensions of E λ ( L) is ( s+ ν)/ λ. When L is the set of all positive integers, the result specializes to a well-known theorem of Jarnik (Math. Zeitschrift 33, 1931, 505–543). It also includes some results of Eggleston (Proc. Lond. Math. Soc. Series 2 54, 1951, 42–93) about arithmetical progressions and sets of positive density (ν=1) and geometrical progressions (ν=0).

On Certain Sets of Positive Density

On Certain Sets of Positive Density