Exceptional sets for Diophantine inequalities

We apply Freeman's variant of the Davenport-Heilbronn method to investigate the exceptional set of real numbers not close to some value of a given real diagonal form at an integral argument. Under appropriate conditions, we show that the exceptional set in the interval [-N,N] has measure O(N^{1-c}), for a positive number c

Near-optimal mean value estimates for multidimensional Weyl sums

We obtain sharp estimates for multidimensional generalisations of Vinogradov's mean value theorem for arbitrary translation-dilation invariant systems, achieving constraints on the number of variables approaching those conjectured to be the best possible. Several applications of our bounds are discussed

On the Distribution of Generating Functions

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/135518/1/blms0113.pd

Number fields and function fields:Coalescences, contrasts and emerging applications

The similarity between the density of the primes and the density of irreducible polynomials defined over a finite field of q elements was first observed by Gauss. Since then, many other analogies have been uncovered between arithmetic in number fields and in function fields defined over a finite field. Although an active area of interaction for the past half century at least, the language and techniques used in analytic number theory and in the function field setting are quite different, and this has frustrated interchanges between the two areas. This situation is currently changing, and there has been substantial progress on a number of problems stimulated by bringing together ideas from each field. We here introduce the papers published in this Theo Murphy meeting issue, where some of the recent developments are explained

Some remarks on Vinogradov's mean value theorem and Tarry's problem

Let W(k, 2) denote the, least number s for which the system of equations has a solution with . We show that for large k one has W(k, 2)≦1/2k 2 (log k +loglog k + O (1)), and moreover that when K is large, one has W(k, 2)≦1/2k(k+1)+1 for at least one value k in the interval [ K, K 3/4+ε ]. We show also that the least s for which the expected asymptotic formula holds for the number of solutions of the above system of equations, inside a box, satisfies s ≦ k 2 (log k + O (loglog k ).Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/41627/1/605_2005_Article_BF01320189.pd

On Pairs of Diagonal Quintic Forms

We demonstrate that a pair of additive quintic equations in at least 34 variables has a nontrivial integral solution, subject only to an 11-adic solubility hypothesis. This is achieved by an application of the Hardy–Littlewood method, for which we require a sharp estimate for a 33.998th moment of quintic exponential sums. We are able to employ p -adic iteration in a form that allows the estimation of such a mean value over a complete unit square, thereby providing an approach that is technically simpler than those of previous workers and flexible enough to be applied to related problems.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/42604/1/10599_2004_Article_334960.pd

Unbiased taxonomic annotation of metagenomic samples

The classification of reads from a metagenomic sample using a reference taxonomy is usually based on first mapping the reads to the reference sequences and then classifying each read at a node under the lowest common ancestor of the candidate sequences in the reference taxonomy with the least classification error. However, this taxonomic annotation can be biased by an imbalanced taxonomy and also by the presence of multiple nodes in the taxonomy with the least classification error for a given read. In this article, we show that the Rand index is a better indicator of classification error than the often used area under thereceiver operating characteristic (ROC) curve andF-measure for both balanced and imbalanced reference taxonomies, and we also address the second source of bias by reducing the taxonomic annotation problem for a whole metagenomic sample to a set cover problem, for which a logarithmic approximation can be obtained in linear time and an exact solution can be obtained by integer linear programming. Experimental results with a proof-of-concept implementation of the set cover approach to taxonomic annotation in a next release of the TANGO software show that the set cover approach further reduces ambiguity in the taxonomic annotation obtained with TANGO without distorting the relative abundance profile of the metagenomic sample.Peer ReviewedPostprint (published version

Scaling metagenome sequence assembly with probabilistic de Bruijn graphs

Deep sequencing has enabled the investigation of a wide range of environmental microbial ecosystems, but the high memory requirements for {\em de novo} assembly of short-read shotgun sequencing data from these complex populations are an increasingly large practical barrier. Here we introduce a memory-efficient graph representation with which we can analyze the k-mer connectivity of metagenomic samples. The graph representation is based on a probabilistic data structure, a Bloom filter, that allows us to efficiently store assembly graphs in as little as 4 bits per k-mer, albeit inexactly. We show that this data structure accurately represents DNA assembly graphs in low memory. We apply this data structure to the problem of partitioning assembly graphs into components as a prelude to assembly, and show that this reduces the overall memory requirements for {\em de novo} assembly of metagenomes. On one soil metagenome assembly, this approach achieves a nearly 40-fold decrease in the maximum memory requirements for assembly. This probabilistic graph representation is a significant theoretical advance in storing assembly graphs and also yields immediate leverage on metagenomic assembly