We formulate a thermodynamical approach to the study of distribution of modular symbols, motivated by the work of Baladi-Vallee. We introduce the modular partitions of continued fractions and observe that the statistics for modular symbols follow from the behavior of modular partitions. We prove the limit Gaussian distribution and residual equidistribution for modular partitions as a vector-valued random variable on the set of rationals whose denominators are up to a fixed positive integer by studying the spectral properties of transfer operator associated to the underlying dynamics. The approach leads to a few applications. We show an average version of conjectures of Mazur-Rubin on statistics for the period integrals of an elliptic newform. We further observe that the equidistribution of mod $p$ values of modular symbols leads to mod $p$ non-vanishing results for special modular $L$-values twisted by a Dirichlet character.

pdf/dynamics-of-continued-fractions-and-distribution-of-modular-ooojvh4dk6.pdf

Dynamics of continued fractions and distribution of modular symbols

Hwang’s quasi-power theorem asserts that a sequence of random variables whose moment generating functions are approximately given by powers of some analytic function is asymptotically normally distributed. This theorem is generalised to higher dimensional random variables. To obtain this result, a higher dimensional analogue of the Berry–Esseen inequality is proved, generalising a two-dimensional version by Sadikova.

/pdf/higher-dimensional-quasi-power-theorem-and-berry-esseen-45ymefc3sr.pdf

Higher dimensional quasi-power theorem and Berry–Esseen inequality

We show that the (morphic) sequence $(-1)^{s_\varphi(n)}$ is asymptotically orthogonal to all bounded multiplicative functions, where $s_\varphi$ denotes the Zeckendorf sum-of-digits function. In particular we have $\sum_{n<N} (-1)^{s_\varphi(n)} \mu(n) = o(N)$, that is, this sequence satisfies the Sarnak conjecture.

Möbius orthogonality for the Zeckendorf sum-of-digits function

Many digital functions studied in the literature, e.g., the summatory function of the base-$k$ sum-of-digits function, have a behavior showing some periodic fluctuation. Such functions are usually studied using techniques from analytic number theory or linear algebra. In this paper we develop a method based on exotic numeration systems and we apply it on two examples motivated by the study of generalized Pascal triangles and binomial coefficients of words.

Behavior of digital sequences through exotic numeration systems

We show that any $q$-multiplicative sequence which is \emph{oscillating} of order $1$, i.e.\ does not correlate with linear phase functions $e^{2\pi i n\alpha}$ ($\alpha \in \mathbb{R})$, is Gowers uniform of all orders, and hence in particular does not correlate with polynomial phase functions $e^{2\pi i p(n)}$ ($p \in \mathbb{R}[x]$). Quantitatively, we show that any $q$-multiplicative sequence which is of Gelfond type of order 1 is automatically of Gelfond type of all orders. Consequently, any such $q$-multiplicative sequence is a good weight for ergodic theorems. We also obtain combinatorial corollaries concerning linear patterns in sets which are described in terms of sums of digits.

On uniformity of q-multiplicative sequences

We study the joint distribution of the input sum and the output sum of a deterministic transducer. Here, the input of this finite-state machine is a uniformly distributed random sequence.We give a simple combinatorial characterization of transducers for which the output sum has bounded variance, and we also provide algebraic and combinatorial characterizations of transducers for which the covariance of input and output sum is bounded, so that the two are asymptotically independent.Our results are illustrated by several examples, such as transducers that count specific blocks in the binary expansion, the transducer that computes the Gray code, or the transducer that computes the Hamming weight of the width- w non-adjacent form digit expansion. The latter two turn out to be examples of asymptotic independence.

Variances and covariances in the Central Limit Theorem for the output of a transducer

As a generalization of the sum of digits function and other digital sequences, sequences defined as the sum of the output of a transducer are asymptotically analyzed The input of the transducer is a random integer in $[0, N)$ Analogues in higher dimensions are also considered Sequences defined by a certain class of recursions can be written in this framework 
Depending on properties of the transducer, the main term, the periodic fluctuation and an error term of the expected value and the variance of this sequence are established The periodic fluctuation of the expected value is H\"older continuous and, in many cases, nowhere differentiable A general formula for the Fourier coefficients of this periodic function is derived Furthermore, it turns out that the sequence is asymptotically normally distributed for many transducers As an example, the abelian complexity function of the paperfolding sequence is analyzed This sequence has recently been studied by Madill and Rampersad

Output sum of transducers: Limiting distribution and periodic fluctuation

In this paper, we introduce the notion of $q$-quasiadditivity of arithmetic functions, as well as the related concept of $q$-quasimultiplicativity, which generalise strong $q$-additivity and -multiplicativity, respectively. We show that there are many natural examples for these concepts, which are characterised by functional equations of the form $f(q^{k+r}a + b) = f(a) + f(b)$ or $f(q^{k+r}a + b) = f(a) f(b)$ for all $b < q^k$ and a fixed parameter $r$. In addition to some elementary properties of $q$-quasiadditive and $q$-quasimultiplicative functions, we prove characterisations of $q$-quasiadditivity and $q$-quasimultiplicativity for the special class of $q$-regular functions. The final main result provides a general central limit theorem that includes both classical and new examples as corollaries.

On $q$-Quasiadditive and $q$-Quasimultiplicative Functions

The new finite state machine package in the mathematics software system SageMath is presented and illustrated by many examples. Several combinatorial problems, in particular digit problems, are introduced, modeled by automata and transducers and solved using SageMath. In particular, we compute the asymptotic Hamming weight of a non-adjacent-form-like digit expansion, which was not known before.

pdf/automata-in-sagemath-combinatorics-meet-theoretical-computer-5k89nx6n5z.pdf

Automata in SageMath---Combinatorics meet Theoretical Computer Science

Handle everyday research tasks with reliable, citation-backed results

Your personal Research Agent to handle research tasks with citation-backed results

Popular Tasks used by Researchers

How can I help with your research?

Meet SciSpace

Get more enhanced response by uploading the PDFs you want me to reference.

No relevant PDFs in your library

SciSpace is the AI research assistant for academics. Run systematic literature reviews on 280M+ papers, and write papers with cited sources. Trusted by 1M+ students, PhDs & researchers.

SciSpace | AI for Research

Analyze PDFs

Code & Manuscripts

Funding & Grants

Literature & Patents

Medical & Clinical Data

Systematic Review

Visualize & Present

Web & Data

Build a Google Scholar-like website for your research.

Build a website

Create charts and images for your research

Create a Chart

Write a paper for submission to a journal

Draft a manuscript

Patent Search

Design eye-catching scientific posters in minutes.

Scientific Poster Generation

Systematic Literature Review

One task is running at the moment. Your messages will be shown right after.

Drag and drop or click here to browse

Loved by <highlight>1 million+</highlight> researchers

Extract a list of specific topics and their sources from unstructured text

Topics

Compare and analyze relevant papers that matches with your search

Papers

Get insights from PDFs and bookmarked papers from your library

My library

Recent searches

Try searching for:

Catch AI-generated content in scholarly and non-scholarly content

{ai} Detector

Ai Writer

Get PDF Summaries, highlighted text explanations 

Chat with PDF

Effortlessly create in-text citations and bibliographies in APA and 2,500 other formats

Citation generator

Get explanations, summaries, and answers on academic papers

Ease up your research workflow with {scispace}'s cohort of exciting AI tools

Elevate your academic writing skills and convey your ideas the way you want

Paraphraser

Explore our range of reading and writing tools

Your file is being prepared and should be ready in a few minutes. If it's a large file, it might take a bit longer. You can close this window, and we'll email you the file when it's done.

You have reached a maximum limit of <strong>{limit}</strong> columns in the table. Remove at least <strong>1</strong> column to add or create another one.

Sara Kropf

Author Tools

Chat about Author