Related papers: The gerrymander sequence, or A348456

Finding the length of the longest increasing subsequence (LIS) is a classic algorithmic problem. Let $n$ denote the size of the array. Simple $O(n\log n)$ algorithms are known for this problem. We develop a polylogarithmic time randomized…

We introduce the notion of {\bf a}-walk $S(n)=a_1 X_1+\dots+a_n X_n$, based on a sequence of positive numbers ${\bf a}=(a_1,a_2,\dots)$ and a Rademacher sequence $X_1,X_2,\dots$. We study recurrence/transience (properly defined) of such…

In two recent works \cite{BMM,BK}, it has been shown that the counting generating functions (CGF) for the 23 walks with small steps confined in a quadrant and associated with a finite group of birational transformations are holonomic, and…

We examine the exponentially improved asymptotic expansion of the Lerch zeta function $L(\lambda,a,s)=\sum_{n=1}^\infty \exp (2\pi ni\lambda)/(n+a)^s$ for large complex values of $a$, with $\lambda$ and $s$ regarded as parameters. It is…

Quantum computing is an exciting non-Von Neumann paradigm, offering provable speedups over classical computing for specific problems. However, the practical limits of classical simulatability for quantum circuits remain unclear, especially…

We investigate the geometry of predictive information across the layers of large language models (LLMs). We repurpose representation lenses-learned affine maps trained to predict the next token from intermediate residual streams-as…

The fundamental question considered in algorithms on strings is that of indexing, that is, preprocessing a given string for specific queries. By now we have a number of efficient solutions for this problem when the queries ask for an exact…

We show that if a Barker sequence of length $n>13$ exists, then either $n=3979201339721749133016171583224100$, or $n > 4\cdot10^{33}$. This improves the lower bound on the length of a long Barker sequence by a factor of nearly 2000. We also…

We revisit the so-called "Three Squares Lemma" by Crochemore and Rytter [Algorithmica 1995] and, using arguments based on Lyndon words, derive a more general variant which considers three overlapping squares which do not necessarily share a…

We consider weighted small step walks in the positive quadrant, and provide algebraicity and differential transcendence results for the underlying generating functions: we prove that depending on the probabilities of allowed steps, certain…

One of the central problems in additive combinatorics is to determine how large a subset of the first $N$ integers can be before it is forced to contain $k$ elements forming an arithmetic progression. Around 25 years ago, Gowers proved the…

We investigate the geometry of a random rational lemniscate $\Gamma$, the level set $\{|r(z)|=1\}$ on the Riemann sphere of the modulus of a random rational function $r$. We assign a probability distribution to the space of rational…

In the present work, we investigate real numbers whose sequence of partial quotients enjoys some combinatorial properties involving the notion of palindrome. We provide three new transendence criteria, that apply to a broad class of…

The nature of the alignment with gaps corresponding to a longest common subsequence (LCS) of two independent iid random sequences drawn from a finite alphabet is investigated. It is shown that such an optimal alignment typically matches…

We advocate for a new paradigm of cosmological likelihood-based inference, leveraging recent developments in machine learning and its underlying technology, to accelerate Bayesian inference in high-dimensional settings. Specifically, we…

Associated to a finite measure on the real line with finite moments are recurrence coefficients in a three-term formula for orthogonal polynomials with respect to this measure. These recurrence coefficients are frequently inputs to modern…

The theory of transient growth describes how linear mechanisms can cause temporary amplification of disturbances even when the linearized system is asymptotically stable as defined by its eigenvalues. This growth is traditionally quantified…

In the context of countable groups of polynomial volume growth, we consider a large class of random walks that are allowed to take long jumps along multiple subgroups according to power law distributions. For such a random walk, we study…

Self-avoiding walks on the body-centered-cubic (BCC) and face-centered-cubic (FCC) lattices are enumerated up to lengths 28 and 24, respectively, using the length-doubling method. Analysis of the enumeration results yields values for the…

We consider the task of privately obtaining prediction error guarantees in ordinary least-squares regression problems with Gaussian covariates (with unknown covariance structure). We provide the first sample-optimal polynomial time…