相关论文: On the level sets of the Takagi-van der Waerden fu…

This paper examines level sets of functions of the form $f(x)=\sum_{n=0}^\infty \frac{r_n}{2^n}\phi(2^n x)$, where phi(x) is the distance from x to the nearest integer, and r_n equals 1 or -1 for each n. Such functions are referred to as…

In this paper, we investigate the Takagi-van der Waerden function, $$ T_r(x) = \sum_{n=0}^{\infty} \frac{\phi(r^n x)}{r^n} ,\quad x\in [0,1], \quad r \in \mathbb{Z}^+, $$ where $\phi(x)={\rm dist}(x,\mathbb{Z})$ represents the distance from…

Let T be Takagi's continuous but nowhere-differentiable function. This paper considers the size of the level sets of T both from a probabilistic point of view and from the perspective of Baire category. We first give more elementary proofs…

We introduce the Takagi--van der Waerden function with parameters $a{>}b{>}0$ by setting $f_{a,b}(x)=\sum\limits_{n=1}^\infty b^n d\big(x,S_n\big)$, where $S_n$ is a maximal $\frac1{a^n}$-separated set in a metric space $X$. So, if…

We study the Takagi-van der Waerden functions $f_r (x)$, a well-known class of continuous but nowhere differentiable functions, from probabilistic point of view. As an application of elephant random walks remembering the very recent past…

The Takagi-van der Waerden functions are a well-known class of continuous but nowhere differentiable functions. In this paper, we study their weighted versions, the Takagi-van der Waerden class functions $f_{r,a}(x)$, from a probabilistic…

The Takagi function {\tau} : [0, 1] \rightarrow [0, 1] is a continuous non-differentiable function constructed by Takagi in 1903. This paper studies the level sets L(y) = {x : {\tau}(x) = y} of the Takagi function {\tau}(x). It shows that…

This paper examines level sets of two families of continuous, nowhere differentiable functions (one a subfamily of the other) defined in terms of the "tent map". The well-known Takagi function is a special case. Sharp upper bounds are given…

The Takagi function \tau : [0, 1] \to [0, 1] is a continuous non-differentiable function constructed by Takagi in 1903. The level sets L(y) = {x : \tau(x) = y} of the Takagi function \tau(x) are studied by introducing a notion of local…

In this paper, we prove that for some Generalized Takagi Classes, in particular for the Takagi-Van der Waerden Class, the functions are nowhere differentiable if, and only if, the sequence of weights does not belong to $c_0$.

A function f:R -> R is approximately continuous iff it is continuous in the density topology, i.e., for any ordinary open set U the set E=f^{-1}(U) is measurable and has Lebesgue density one at each of its points. Denjoy proved that…

Let $F \subset \R^n$ be a closed set and $n=2$ or $n=3$. S. Ferry (1975) proved that then, for almost all $r>0$, the level set (distance sphere, $r$-boundary) $S_r(F):= \{x \in \R^n: \dist(x,F) = r\}$ is a topological $(n-1)$-dimensional…

In this short note, we show that the distance function to any finite set $X\subset \mathbb{R}^n$ is a topological Morse function, regardless of whether $X$ is in general position. We also precisely characterize its topological critical…

We study one class of continuous functions $f$ defined on segment $[0,1]$ by equality $$ f(x)=\delta_{\alpha_1(x)1}+\sum^{\infty}_{k=2}\left[\delta_{\alpha_k(x)k}\prod^{k-1}_{j=1}g_{\alpha_j…

Given a continuous function $f:[a,b]\to\mathbb{R}$ such that $f(a)=f(b)$, we investigate the set of distances $|x-y|$ where $f(x)=f(y)$. In particular, we show that the only distances this set must contain are ones which evenly divide…

Let T be Takagi's continuous but nowhere-differentiable function. It is known that almost all level sets (with respect to Lebesgue measure on the range of T) are finite. We show that the most common cardinality of the level sets of T is…

Functions with uniform level sets can represent orders, preference relations or other binary relations and thus turn out to be a tool for scalarization that can be used, e.g., in multicriteria optimization, decision theory, mathematical…

For $x\in [0,1],$ the run-length function $r_n(x)$ is defined as the length of the longest run of $1$'s amongst the first $n$ dyadic digits in the dyadic expansion of $x.$ Erd\H{o}s and R\'enyi proved that…

Let $\beta > 1$ and the run-length function $r_n(x,\beta)$ be the maximal length of consecutive zeros amongst the first n digits in the $\beta$-expansion of $x\in[0,1]$. The exceptional set $$E_{\max}^{\varphi}=\left\{x \in…

The functions of the Takagi exponential class are similar in construction to the continuous, nowhere differentiable Takagi function described in 1901. They have one real parameter $v\in (-1;1)$ and at points $x\in{\mathbb R}$ are defined by…