arXiv subjects

Karl Mahlburg

Publications and source records attributed to Karl Mahlburg.

At least 19 recordsLinked to original sources

arXiv2020

On a Tauberian Theorem of Ingham and Euler-Maclaurin Summation

We discuss two theorems in analytic number theory and combinatory analysis that have seen increased use in recent years. A corollary to a Tauberian theorem of Ingham allows one to quickly prove asymptotic formulas for arithmetic sequences, so long as the corresponding generating function exhibits exponential growth of a certain form near its radius of convergence. Two common methods for proving the required analytic behavior are modular transformations and Euler-Maclaurin summation. However, these results are sometimes stated without certain technical conditions that are necessary for the complex analytic techniques that appear in Ingham's proof. We carefully examine the precise statements and proofs of these results, and find that in practice, the technical conditions are satisfied for those cases appearing in recent applications. We also generalize the classical approach of Euler-Maclaurin summation in order to prove asymptotic expansions for series with complex values, simple poles, or multi-dimensional summation indices.

math.NT↗

arXiv2019

The Asymptotic Distribution of the Rank for Unimodal Sequences

We study the asymptotic behavior of the rank statistic for unimodal sequences. We use analytic techniques involving asymptotic expansions in order to prove asymptotic formulas for the moments of the rank. Furthermore, when appropriately normalized, the values of the unimodal rank asymptotically follow a logistic distribution. We also prove similar results for Durfee unimodal sequences and semi-strict unimodal sequences, with the only major difference being that the (normalized) rank for semistrict unimodal sequences has a distributional limit of a point mass probability distribution.

math.NT↗

arXiv2019

Quantum modular forms and plumbing graphs of 3-manifolds

In this paper, we study quantum modular forms in connection to quantum invariants of plumbed 3-manifolds introduced recently by Gukov, Pei, Putrov, and Vafa. We explicitly compute these invariants for any $3$-leg star plumbing graphs whose associated matrix is unimodular and positive definite. For these graphs we confirm a quantum modularity conjecture of Gukov. We also analyze the invariants for general $n$-leg star graphs with unimodular plumbing matrices, and prove that they can be expressed as linear combinations of quantum modular forms.

math.QA↗

arXiv2019

Higher depth quantum modular forms and plumbed $3$-manifolds

In this paper we study new invariants $\widehat{Z}_{\boldsymbol{a}}(q)$ attached to plumbed $3$-manifolds that were introduced by Gukov, Pei, Putrov, and Vafa. These remarkable $q$-series at radial limits conjecturally compute WRT invariants of the corresponding plumbed $3$-manifold. Here we investigate the series $\widehat{Z}_{0}(q)$ for unimodular plumbing ${\tt H}$-graphs with six vertices. We prove that for every positive definite unimodular plumbing matrix, $\widehat{Z}_{0}(q)$ is a depth two quantum modular form on $\mathbb{Q}$.

math.NT↗

arXiv2019

Proofs and reductions of various conjectured partition identities of Kanade and Russell

We prove seven of the Rogers-Ramanujan type identities modulo $12$ that were conjectured by Kanade and Russell. Included among these seven are the two original modulo $12$ identities, in which the products have asymmetric congruence conditions, as well as the three symmetric identities related to the principally specialized characters of certain level $2$ modules of $A_9^{(2)}$. We also give reductions of four other conjectures in terms of single-sum basic hypergeometric series.

math.NT↗

arXiv2018

Peak positions of strongly unimodal sequences

We study combinatorial and asymptotic properties of the rank of strongly unimodal sequences. We find a generating function for the rank enumeration function, and give a new combinatorial interpretation of the ospt-function introduced by Andrews, Chan, and Kim. We conjecture that the enumeration function for the number of unimodal sequences of a fixed size and varying rank is log-concave, and prove an asymptotic result in support of this conjecture. Finally, we determine the asymptotic behavior of the rank for strongly unimodal sequences, and prove that its values (when appropriately renormalized) are normally distributed with mean zero in the asymptotic limit.

math.NT↗

arXiv2018

On characters of $L_{\frak{sl}_n}(-Λ_0)$-modules

We use recent results of Rolen, Zwegers, and the first author to study characters of irreducible (highest weight) modules for the vertex operator algebra $L_{\frak{sl}_\ell}(-Λ_0)$. We establish asymptotic behaviors of characters for the (ordinary) irreducible $L_{\frak{sl}_\ell}(-Λ_0)$-modules. As a consequence we prove that their quantum dimensions are one, as predicted by representation theory. We also establish a full asymptotic expansion of irreducible characters for $\frak{sl}_3$. Finally, we determine a decomposition formula for the full characters in terms of unary theta and false theta functions which allows us to study their modular properties.

math.NT↗

arXiv2017

On a modularity conjecture of Andrews, Dixit, Schultz, and Yee for a variation of Ramanujan's $ω(q)$

We analyze the mock modular behavior of $\bar{P}_ω(q)$, a partition function introduced by Andrews, Dixit, Schultz, and Yee. This function arose in a study of smallest parts functions related to classical third order mock theta functions, one of which is $ω(q)$. We find that the modular completion of $\bar{P}_ω(q)$ is not simply a harmonic Maass form, but is instead the derivative of a linear combination of products of various harmonic Maass forms and theta functions. We precisely describe its behavior under modular transformations and find that the image under the Maass lowering operator lies in a relatively simpler space.

math.NT↗

arXiv2015

Approximate polynomial structure in additively large sets

We show that any subset of the natural numbers with positive logarithmic Banach density contains a set that is within a factor of two of a geometric progression, improving the bound on a previous result of the authors. Density conditions on subsets of the natural numbers that imply the existence of approximate powers of arithmetic progressions are developed and explored.

math.CO↗

arXiv2015

High density piecewise syndeticity of product sets in amenable groups

M. Beiglböck, V. Bergelson, and A. Fish proved that if $G$ is a countable amenable group and $A$ and $B$ are subsets of $G$ with positive Banach density, then the product set $AB$ is piecewise syndetic. This means that there is a finite subset $E$ of $G$ such that $EAB$ is thick, that is, $EAB$ contains translates of any finite subset of $G$. When $G=\mathbb{Z}$, this was first proven by R. Jin. We prove a quantitative version of the aforementioned result by providing a lower bound on the density (with respect to a Følner sequence) of the set of witnesses to the thickness of $% EAB$. When $G=\mathbb{Z}^d$, this result was first proven by the current set of authors using completely different techniques.

math.CO↗

arXiv2015

A monad measure space for logarithmic density

We provide a framework for proofs of structural theorems about sets with positive Banach logarithmic density. For example, we prove that if $A\subseteq \mathbb{N}$ has positive Banach logarithmic density, then $A$ contains an approximate geometric progression of any length. We also prove that if $A,B\subseteq \mathbb{N}$ have positive Banach logarithmic density, then there are arbitrarily long intervals whose gaps on $A\cdot B$ are multiplicatively bounded, a multiplicative version Jin's sumset theorem. The main technical tool is the use of a quotient of a Loeb measure space with respect to a multiplicative cut.

math.LO↗

arXiv2015

False theta functions and companions to Capparelli's identities

Capparelli conjectured two modular identities for partitions whose parts satisfy certain gap conditions, where were motivated by the calculation of characters for the standard modules of certain affine Lie algebras and by vertex operator theory. These identities were subsequently proved and refined by Andrews, who related them to Jacobi theta functions, and also by Alladi-Andrews-Gordon, Capparelli, and Tamba-Xie. In this paper we prove two new companions to Capparelli's identities, where the evaluations are expressed in terms of Jacobi theta functions and false theta functions.

math.NT↗

arXiv2015

Distinct parts partitions without sequences

Partitions without sequences of consecutive integers as parts have been studied recently by many authors, including Andrews, Holroyd, Liggett, and Romik, among others. Their results include a description of combinatorial properties, hypergeometric representations for the generating functions, and asymptotic formulas for the enumeration functions. We complete a similar investigation of partitions into distinct parts without sequences, which are of particular interest due to their relationship with the Rogers-Ramanujan identities. Our main results include a double series representation for the generating function, an asymptotic formula for the enumeration function, and several combinatorial inequalities.

math.NT↗

arXiv2014

Double series representations for Schur's partition function and related identities

We prove new double summation hypergeometric $q$-series representations for several families of partitions, including those that appear in the famous product identities of Göllnitz, Gordon, and Schur. We give several different proofs for our results, using bijective partitions mappings and modular diagrams, the theory of $q$-difference equations and recurrences, and the theories of summation and transformation for $q$-series. We also consider a general family of similar double series and highlight a number of other interesting special cases.

math.NT↗

arXiv2014

On a sumset conjecture of Erdős

Erdős conjectured that for any set $A\subseteq \mathbb{N}$ with positive lower asymptotic density, there are infinite sets $B,C\subseteq \mathbb{N}$ such that $B+C\subseteq A$. We verify Erdős' conjecture in the case that $A$ has Banach density exceeding $\frac{1}{2}$. As a consequence, we prove that, for $A\subseteq \mathbb{N}$ with positive Banach density (a much weaker assumption than positive lower density), we can find infinite $B,C\subseteq \mathbb{N}$ such that $B+C$ is contained in the union of $A$ and a translate of $A$. Both of the aforementioned results are generalized to arbitrary countable amenable groups. We also provide a positive solution to Erdős' conjecture for subsets of the natural numbers that are pseudorandom.

math.NT↗

arXiv2013

A partition identity and the universal mock theta function $g_2$

We prove analytic and combinatorial identities reminiscent of Schur's classical partition theorem. Specifically, we show that certain families of overpartitions whose parts satisfy gap conditions are equinumerous with partitions whose parts satisfy congruence conditions. Furthermore, if small parts are excluded, the resulting overpartitions are generated by the product of a modular form and Gordon and McIntosh's universal mock theta function. Finally, we give an interpretation for the universal mock theta function at real arguments in terms of certain conditional probabilities.

math.NT↗

arXiv2013

High density piecewise syndeticity of sumsets

Renling Jin proved that if A and B are two subsets of the natural numbers with positive Banach density, then A+B is piecewise syndetic. In this paper, we prove that, under various assumptions on positive lower or upper densities of A and B, there is a high density set of witnesses to the piecewise syndeticity of A+B. Most of the result are shown to hold more generally for subsets of Z^d. The key technical tool is a Lebesgue density theorem for measure spaces induced by cuts in the nonstandard integers.

math.NT↗

arXiv2013

Asymptotic formulas for stacks and unimodal sequences

We study enumeration functions for unimodal sequences of positive integers, where the size of a sequence is the sum of its terms. We survey known results for a number of natural variants of unimodal sequences, including Auluck's generalized Ferrer diagrams, Wright's stacks, and Andrews' convex compositions. These results describe combinatorial properties, generating functions, and asymptotic formulas for the enumeration functions. We also prove several new asymptotic results that fill in the notable missing cases from the literature, including an open problem in statistical mechanics due to Temperley. Furthermore, we explain the combinatorial and asymptotic relationship between partitions, Andrews' Frobenius symbols, and stacks with summits.

math.NT↗