Poissonian pair correlation and discrepancy

A sequence ${\left(x_n\right)}_{n=1}^{\infty }$ on the torus $\mathbb{T}?[0,1]$ is said to exhibit Poissonian pair correlation if the local gaps behave like the spacings of a Poisson random variable, i.e.

$\underset{N\to \infty }{lim}\frac{1}{N}\#\left\{1\le m\ne n\le N:|x_m?x_n|\le \frac{s}{N}\right\}=2s\text{almost surely.}$

We show that being close to Poissonian pair correlation for few values of $s$ is enough to deduce global regularity statements: if, for some $0<\delta <1∕2$, a set of points $\left\{x_1,\dots ,x_N\right\}$ satisfies

$\frac{1}{N}\#\left\{1\le m\ne n\le N:|x_m?x_n|\le \frac{s}{N}\right\}\le (1+\delta )2s\text{for all}1\le s\le (8∕\delta )\sqrt{logN},$

then the discrepancy $D_N$ of the set satisfies $D_N?{\delta }^{1∕3}+N^{?1∕3}{\delta }^{?1∕2}$. We also show that distribution properties are reflected in the global deviation from the Poissonian pair correlation

$N^2{D}_N^5?\frac{2}{N}{\int }_0^{N∕2}{\left|\frac{1}{N}\#\left\{1\le m\ne n\le N:|x_m?x_n|\le \frac{s}{N}\right\}?2s\right|}^{2}ds?N^2{D}_N^2,$

where the lower bound is conditioned on $D_N?N^{?1∕3}$. The proofs use a connection between exponential sums, the heat kernel on $\mathbb{T}$ and spatial localization. Exponential sum estimates are obtained as a byproduct. We also describe a connection to diaphony and several open problems.     

Low discrepancy sequences failing Poissonian pair correlations

Let G be a locally compact totally disconnected topological group. Under a necessary mild assumption, we show that the irreducible unitary representations of G are uniformly admissible if and only if the irreducible smooth representations of G are uniformly admissible. An analogous result for *-algebras is also established. We further show that the property of having uniformly admissible irreducible smooth representations is inherited by finite-index subgroups and overgroups of G.     

模糊信息差异及唯一性定理证明

定义了模糊相对熵,基于模糊熵和距离定义了模糊信息差异、拟模糊信息差异,并讨论了模糊信息差异唯一性定理以及模糊信息差异的性质.     

On low discrepancy sequences and low discrepancy ergodic transformations of the multidimensional unit cube

In this paper we describe a third class of low discrepancy sequences. Using a lattice Γ ? ? ^s , we construct Kronecker-like and van der Corput-like ergodic transformations T _1,Γ and T _2,Γ of [0, 1) ^s . We prove that for admissible lattices Γ, (T _ν,Γ ⁿ (x))_n≥0 is a low discrepancy sequence for all x ∈ [0, 1) ^s and ν ∈ {1, 2}. We also prove that for an arbitrary polyhedron P ? [0, 1) ^s , for almost all lattices Γ ∈ L _s = SL(s,?)/SL(s, ?) (in the sense of the invariant measure on L _s ), the following asymptotic formula

$\# \{ 0 \le n < N:T_{v,\Gamma }^n(x) \in P\} = NvolP + O({(\ln N)^{s + \varepsilon }}),N \to \infty$

holds with arbitrary small ? > 0, for all x ∈ [0, 1) ^s , and ν ∈ {1, 2}.

     

On the discrepancy of 3 permutations

Consider different orderings of an n-element set and the hypergraph consisting of the intervals of these orderings. A conjecture of J. Beck states that in the case of three orderings this hypergraph has bounded discrepancy. We show that for any constant number of orderings the discrepancy is O(log n). the proof also gives an efficient algorithm to determine such a coloring.     

On one-sided torsion pair

Motivated by the concept of a torsion pair in a pre-triangulated category induced by Beligiannis and Reiten, the notion of a left (right) torsion pair in the left (right) triangulated category is introduced and investigated. We provide new connections between different aspects of torsion pairs in one-sided triangulated categories, pre-triangulated categories, stable categories and derived categories.     

On the discrepancy of convex plane sets

LetD ₂ (N) be the discrepancy function of the class of convex sets in the unit square [0, 1)² as defined in the introduction. A well known result of W. M. Schmidt states thatD ₂ f(N)>constN ^1/3. In this paper it is shown that Schmidt's bound is nearly best possible, more precisely, $$D_2 (N)< const N^{1/3} (\log N)^4 .$$      

On pair correlations and Hausdorff dimension

We consider a system of “generalised linear forms” defined at a point x = (x _{(i, j)}) in a subset of R ^d by

On the discrepancy between two Zagreb indices

We examine the quantity

$S\left(G\right)=\sum _{uv\in E\left(G\right)}min(degu,degv)$

over sets of graphs with a fixed number of edges. The main result shows the maximum possible value of $S\left(G\right)$ is achieved by three different classes of constructions, depending on the distance between the number of edges and the nearest triangular number. Furthermore we determine the maximum possible value when the set of graphs is restricted to be bipartite, a forest, or to be planar given sufficiently many edges. The quantity $S\left(G\right)$ corresponds to the difference between two well studied indices, the irregularity of a graph and the sum of the squares of the degrees in a graph. These are known as the first and third Zagreb indices in the area of mathematical chemistry.     

On the discrepancy of some special sequences

We obtain estimates for the discrepancy of the sequence (xs^(d)(q;n))_n=0^∞, where s^(d)(q;n) denotes the sum of the dth powers of the q-ary digits of the nonnegative integer n and x is an irrational number of finite approximation type. Furthermore metric results for a similar type of sequences are given.     

On the discrepancy of (0,1)-sequences

We give bounds for the L^p-discrepancy, , of the van der Corput sequence in base 2. Further, we give a best possible upper bound for the star discrepancy of (0,1)-sequences and show that this bound is attained for the van der Corput sequence. Finally, we give a (0,1)-sequence with essentially smaller star discrepancy than for the van der Corput sequence.     

Explicit relations between pair correlation of zeros and primes in short intervals

In this paper we obtain a quantitative version of the well known theorem by Goldston and Montgomery about the equivalence between the asymptotic behaviors of the mean-square of primes in short intervals, and of the pair-correlation function of the zeros of the Riemann zeta function.     

The pair correlation of zeros of DirichletL-functions and primes in arithmetic progressions

We define a function which correlates the zeros of two DirichletL-functions to the modulusq and we prove an asymptotic estimate for averages of the pair correlation functions over all pairs of characters to (modq). An analogue of Montgomery’s pair correlation conjecture is formulated as to how this estimate can be extended to a greater domain for the parameters that are involved. Based on this conjecture we obtain results about the distribution of primes in an arithmetic progression (to a prime modulusq) and gaps between such primes.     

On the order of discrepancy of the Smolyak grid

Mathematical Notes -     

On the discrepancy of random matrices with many columns

Motivated by the Komlós conjecture in combinatorial discrepancy, we study the discrepancy of random matrices with m rows and n independent columns drawn from a bounded lattice random variable. We prove that for n at least polynomial in m, with high probability the ?_∞‐discrepancy is at most twice the ?_∞‐covering radius of the integer span of the support of the random variable. Applying this result to random t‐sparse matrices, that is, uniformly random matrices with t ones and m?t zeroes in each column, we show that the ?_∞‐discrepancy is at most 2 with probability for . This improves on a bound proved by Ezra and Lovett showing the same bound for n at least m^t.