On an application of almost increasing sequences

In the present paper, a general theorem on ${mid{bar{N},p_n;delta}mid}_k$ summability factors of infinite series has been proved under weaker conditions. Some new results have also been obtained dealing with ${mid{bar{N},p_n }mid}_k$ and ${mid{C,1;delta}mid}_k$ summability factors

On an application of almost increasing sequences

Using an almost increasing sequence, a result of Mazhar (1977) on |C,1|k summability factors has been generalized for |C,α;β|k and |N¯,pn;β|k summability factors under weaker conditions

On higher dimensional Poissonian pair correlation

In this article we study the pair correlation statistic for higher dimensional sequences. We show that for any $d\geq 2$, strictly increasing sequences $(a_n^{(1)}),\ldots, (a_n^{(d)})$ of natural numbers have metric Poissonian pair correlation with respect to sup-norm if their joint additive energy is $O(N^{3-\delta})$ for any $\delta>0$. Further, in two dimension, we establish an analogous result with respect to $2$-norm. As a consequence, it follows that $(\{n\alpha\}, \{n^2\beta\})$ and $(\{n\alpha\}, \{[n\log^An]\beta\})$ ($A \in [1,2]$) have Poissonian pair correlation for almost all $(\alpha,\beta)\in \mathbb{R}^2$ with respect to sup-norm and $2$-norm. This gives a negative answer to the question raised by Hofer and Kaltenb\"ock [15]. The proof uses estimates for 'Generalized' GCD-sums.Comment: Added references and corrected typos. To appear in Journal of Mathematical Analysis and Application

On the distribution of sequences of the form (qny)(q_ny)

We study the distribution of sequences of the form $(q_ny)_{n=1}^\infty$, where $(q_n)_{n=1}^\infty$ is some increasing sequence of integers. In particular, we study the Lebesgue measure and find bounds on the Hausdorff dimension of the set of points $\gamma \in [0,1)$ which are well approximated by points in the sequence $(q_ny)_{n=1}^\infty$. The bounds on Hausdorff dimension are valid for almost every $y$ in the support of a measure of positive Fourier dimension. When the required rate of approximation is very good or if our sequence is sufficiently rapidly growing, our dimension bounds are sharp. If the measure of positive Fourier dimension is itself Lebesgue measure, our measure bounds are also sharp for a very large class of sequences. We also give an application to inhomogeneous Littlewood type problems.Comment: 15 pages. Niclas Technau pointed out to us that Theorems 1 and 2 in the original version were in fact consequences of arXiv:2307.14871 . In this revised version, we have strengthened both theorems to cover a wider class of sequence

Metric number theory, lacunary series and systems of dilated functions

By a classical result of Weyl, for any increasing sequence $(n_k)_{k \geq 1}$ of integers the sequence of fractional parts $(\{n_k x\})_{k \geq 1}$ is uniformly distributed modulo 1 for almost all $x \in [0,1]$. Except for a few special cases, e.g. when $n_k=k, k \geq 1$, the exceptional set cannot be described explicitly. The exact asymptotic order of the discrepancy of $(\{n_k x\})_{k \geq 1}$ is only known in a few special cases, for example when $(n_k)_{k \geq 1}$ is a (Hadamard) lacunary sequence, that is when $n_{k+1}/n_k \geq q > 1, k \geq 1$. In this case of quickly increasing $(n_k)_{k \geq 1}$ the system $(\{n_k x\})_{k \geq 1}$ (or, more general, $(f(n_k x))_{k \geq 1}$ for a 1-periodic function $f$) shows many asymptotic properties which are typical for the behavior of systems of \emph{independent} random variables. Precise results depend on a fascinating interplay between analytic, probabilistic and number-theoretic phenomena. Without any growth conditions on $(n_k)_{k \geq 1}$ the situation becomes much more complicated, and the system $(f(n_k x))_{k \geq 1}$ will typically fail to satisfy probabilistic limit theorems. An important problem which remains is to study the almost everywhere convergence of series $\sum_{k=1}^\infty c_k f(k x)$, which is closely related to finding upper bounds for maximal $L^2$-norms of the form $$\int_0^1 (\max_{1 \leq M \leq N}| \sum_{k=1}^M c_k f(kx)|^2 dx.$$ The most striking example of this connection is the equivalence of the Carleson convergence theorem and the Carleson--Hunt inequality for maximal partial sums of Fourier series. For general functions $f$ this is a very difficult problem, which is related to finding upper bounds for certain sums involving greatest common divisors.Comment: Survey paper for the RICAM workshop on "Uniform Distribution and Quasi-Monte Carlo Methods", held from October 14-18, 2013, in Linz, Austria. This article will appear in the proceedings volume for this workshop, published as part of the "Radon Series on Computational and Applied Mathematics" by DeGruyte

SEED: efficient clustering of next-generation sequences.

MotivationSimilarity clustering of next-generation sequences (NGS) is an important computational problem to study the population sizes of DNA/RNA molecules and to reduce the redundancies in NGS data. Currently, most sequence clustering algorithms are limited by their speed and scalability, and thus cannot handle data with tens of millions of reads.ResultsHere, we introduce SEED-an efficient algorithm for clustering very large NGS sets. It joins sequences into clusters that can differ by up to three mismatches and three overhanging residues from their virtual center. It is based on a modified spaced seed method, called block spaced seeds. Its clustering component operates on the hash tables by first identifying virtual center sequences and then finding all their neighboring sequences that meet the similarity parameters. SEED can cluster 100 million short read sequences in <4 h with a linear time and memory performance. When using SEED as a preprocessing tool on genome/transcriptome assembly data, it was able to reduce the time and memory requirements of the Velvet/Oasis assembler for the datasets used in this study by 60-85% and 21-41%, respectively. In addition, the assemblies contained longer contigs than non-preprocessed data as indicated by 12-27% larger N50 values. Compared with other clustering tools, SEED showed the best performance in generating clusters of NGS data similar to true cluster results with a 2- to 10-fold better time performance. While most of SEED's utilities fall into the preprocessing area of NGS data, our tests also demonstrate its efficiency as stand-alone tool for discovering clusters of small RNA sequences in NGS data from unsequenced organisms.AvailabilityThe SEED software can be downloaded for free from this site: http://manuals.bioinformatics.ucr.edu/home/[email protected] informationSupplementary data are available at Bioinformatics online