Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2408.01864

On the central ball in a translation invariant involutive field

Cristian Cobeli·Aaditya Raghavan·Alexandru Zaharescu
View original source ↗Read source document ↗Export metadata ↓

Abstract

The iterated composition of two operators, both of which are involutions and translation invariant, partitions the set of lattice points in the plane into an infinite sequence of discrete parabolas. Each such parabola contains an associated stairway-like path connecting certain points on it, induced by the alternating application of the aforementioned operators. Any two lattice points in the plane can be connected by paths along the square grid composed of steps either on these stairways or towards taxicab neighbors. This leads to the notion of the parabolic-taxicab distance between two lattice points, obtained as the minimum number of steps of this kind needed to reach one point from the other. In this paper, we describe patterns generated by points on paths of bounded parabolic-taxicab length and provide a complete description of the balls centered at the origin. In particular, we prove an earlier conjecture on the area of these balls.

Explore related subjects

math.NTmath.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Cristian Cobeli, Aaditya Raghavan, Alexandru Zaharescu. 2025-01-06. On the central ball in a translation invariant involutive field. https://arxiv.org/abs/2408.01864

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Asymptotic density of k-almost primes

Landau's well known asymptotic formula $$N_k(x):=\ \mid\{n\leq x : Ω(n)=k\}\mid \ \sim \left( \frac{x}{\log x} \right) \frac{(\log\log x)^{k-1}}{(k - 1)!}\ \ (x \rightarrow \infty),$$ which also holds for $$π_k(x):=\ \mid\{n\leq x : ω(n)=k\}\mid,$$ is known to be fairly poor for $k > 1$, and when $k$ is allowed to tend to infinity with $x$, the study of $N_k(x)$ and $π_k(x)$ becomes very technical [1, Chapter II.6, $§$ 6.1, p.200]. I hope to show that the method described below provides not only a more accurate approach, but rather increases in its asymptotic accuracy as $k$ tends to infinity.

math.NT↗
arXiv2026

Transcendence Meets Normality: Construction of Transcendentally Normal Numbers

In this work, we study real numbers $x$ for which $p(x)$ is (absolutely) normal for every non-constant integer-valued polynomial $p$. We call such numbers transcendentally normal. We prove that almost every real number is transcendentally normal and provide an explicit construction of such a number, based on Sierpinski's covering method and novel ideas involving the so-called stretch function. In the next step, we transform this construction into an algorithm that computes the digits of a t-normal number recursively in all integer bases. Moreover, we extend our covering approach to construct and compute LIL-normal numbers whose discrepancies are of the order predicted by the law of the iterated logarithm. We also take the opportunity to discuss several interesting open problems.

math.NT↗