Related papers: A strict non-standard inequality .999... < 1
We propose a novel foundation for calculus that focuses on the notion of approximations while avoiding the use of limits altogether. Continuity is defined as approximation at a point, while differentiability is defined as approximation with…
According to a popular belief, the decimal digits of mathematical constants such as {\pi} behave like statistically independent random variables, each taking the values 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 with equal probability of 1/10. If…
In this paper, we compute the asymptotic average of the decimals of some real numbers. With the help of this computation, we prove that if a real number cannot be represented as a finite decimal and the asymptotic average of its decimals is…
It is well-known that the notion of limit in the sharp topology of sequences of Colombeau generalized numbers $\widetilde{\mathbb{R}}$ does not generalize classical results. E.g.~the sequence $\frac{1}{n}\not\to0$ and a sequence…
This is a survey of results on definability and undefinability in models of arithmetic. The goal is to present a stark difference between undefinability results in the standard model and much stronger versions about expansions of…
Linear and semidefinite programming (LP, SDP), regularisation through basis pursuit (BP) and Lasso have seen great success in mathematics, statistics, data science, computer-assisted proofs and learning. The success of LP is traditionally…
Simultaneous Diophantine approximation is concerned with the approximation of a point $\mathbf x\in\mathbb R^d$ by points $\mathbf r\in\mathbb Q^d$, with a view towards jointly minimizing the quantities $\|\mathbf x - \mathbf r\|$ and…
A new number system, the set of the non-Dedekindian numbers, is introduced and characterized axiomatically. It is then proved that any hypercontinous hyperreal number system is strictly included in the set of the Non-Dedekindian Numbers.…
In this article, we will introduce methods of non-standard analysis into projective geometry. Especially, we will analyze the properties of a projective space over a non-Archimedean field. Non-Archimedean fields contain numbers that are…
Recently, Harrington, Litman, and Wong [Bulletin of the Australian Mathematical Society, 2024; arXiv:2303.06534] proved that every arithmetic progression contains infinitely many base-$b$ Niven numbers, for any fixed $b\ge 2$. We use a…
We establish sharp bounds for the Hausdorff dimension of sets of irrational numbers in $(0,1)$ whose digits in the $N$-expansion are either uniformly bounded or tend to infinity. For sets with digits bounded by an integer $M \ge N$, we…
Let s and k be integers with s \geq 2 and k \geq 2. Let g_k^{(s)}(n) denote the cardinality of the largest subset of the set {1,2,..., n} that contains no geometric progression of length k whose common ratio is a power of s. Let r_k(\ell)…
Let $b$ be a numeration base. A $b$-Niven number is one that is divisible by the sum of its base $b$ digits. We introduce high degree $b$-Niven numbers. These are $b$-Niven numbers that have a power greater than $1$ that is $b$-Niven…
Consider a non-standard numeration system like the one built over the Fibonacci sequence where nonnegative integers are represented by words over $\{0,1\}$ without two consecutive 1. Given a set $X$ of integers such that the language of…
This article seeks to encourage a mathematical dialog regarding a possible solution to Beals Conjecture. It breaks down one of the worlds most difficult math problems into laymans terms and encourages people to question some of the most…
This paper studies classification with an abstention option in the online setting. In this setting, examples arrive sequentially, the learner is given a hypothesis class $\mathcal H$, and the goal of the learner is to either predict a label…
It is known that Szpiro's conjecture, or equivalently the ABC-conjecture, implies Lang's conjecture giving a uniform lower bound for the canonical height of nontorsion points on elliptic curves. In this note we show that a significantly…
We give a level one result for the "symmetry integral", say $I_f(N,h)$, of essentially bounded $f:\N \to \R$; i.e., we get a kind of "square-root cancellation" \thinspace bound for the mean-square (in $N<x\le 2N$) of the "symmetry"…
The well known binary and decimal representations of the integers, and other similar number systems, admit many generalisations. Here, we investigate whether still every integer could have a finite expansion on a given integer base b, when…
In order to prove irrationality of \sqrt{2} by using only decimal expansions (and not fractions), we develop in detail a model of real numbers based on infinite decimals and arithmetic operations with them.