Restrictions of Brownian motion

Let $\{ B(t) \colon 0\leq t\leq 1\}$ be a linear Brownian motion and let $\dim$ denote the Hausdorff dimension. Let $\alpha>\frac12$ and $1\leq \beta \leq 2$. We prove that, almost surely, there exists no set $A\subset[0,1]$ such that $\dim A>\frac12$ and $B\colon A\to\mathbb{R}$ is $\alpha$-H\"older continuous. The proof is an application of Kaufman's dimension doubling theorem. As a corollary of the above theorem, we show that, almost surely, there exists no set $A\subset[0,1]$ such that $\dim A>\frac{\beta}{2}$ and $B\colon A\to\mathbb{R}$ has finite $\beta$-variation. The zero set of $B$ and a deterministic construction witness that the above theorems give the optimal dimensions.Comment: 6 page

Continuous horizontally rigid functions of two variables are affine

Cain, Clark and Rose defined a function f\colon \RR^n \to \RR to be \emph{vertically rigid} if \graph(cf) is isometric to \graph (f) for every $c \neq 0$. It is \emph{horizontally rigid} if \graph(f(c \vec{x})) is isometric to \graph (f) for every $c \neq 0$ (see \cite{CCR}). In an earlier paper the authors of the present paper settled Jankovi\'c's conjecture by showing that a continuous function of one variable is vertically rigid if and only if it is of the form $a+bx$ or $a+be^{kx}$ (a,b,k \in \RR). Later they proved that a continuous function of two variables is vertically rigid if and only if after a suitable rotation around the z-axis it is of the form $a + bx + dy$, $a + s(y)e^{kx}$ or $a + be^{kx} + dy$ (a,b,d,k \in \RR, $k \neq 0$, s : \RR \to \RR continuous). The problem remained open in higher dimensions. The characterization in the case of horizontal rigidity is surprisingly simpler. C. Richter proved that a continuous function of one variable is horizontally rigid if and only if it is of the form $a+bx$ (a,b\in \RR). The goal of the present paper is to prove that a continuous function of two variables is horizontally rigid if and only if it is of the form $a + bx + dy$ (a,b,d \in \RR). This problem also remains open in higher dimensions. The main new ingredient of the present paper is the use of functional equations

Bruckner--Garg-type results with respect to Haar null sets in C[0,1]C[0,1]

A set $\mathcal{A}\subset C[0,1]$ is \emph{shy} or \emph{Haar null } (in the sense of Christensen) if there exists a Borel set $\mathcal{B}\subset C[0,1]$ and a Borel probability measure $\mu$ on $C[0,1]$ such that $\mathcal{A}\subset \mathcal{B}$ and $\mu\left(\mathcal{B}+f\right) = 0$ for all $f \in C[0,1]$. The complement of a shy set is called a \emph{prevalent} set. We say that a set is \emph{Haar ambivalent} if it is neither shy nor prevalent. The main goal of the paper is to answer the following question: What can we say about the topological properties of the level sets of the prevalent/non-shy many $f\in C[0,1]$? The classical Bruckner--Garg Theorem characterizes the level sets of the generic (in the sense of Baire category) $f\in C[0,1]$ from the topological point of view. We prove that the functions $f\in C[0,1]$ for which the same characterization holds form a Haar ambivalent set. In an earlier paper we proved that the functions $f\in C[0,1]$ for which positively many level sets with respect to the Lebesgue measure $\lambda$ are singletons form a non-shy set in $C[0,1]$. The above result yields that this set is actually Haar ambivalent. Now we prove that the functions $f\in C[0,1]$ for which positively many level sets with respect to the occupation measure $\lambda\circ f^{-1}$ are not perfect form a Haar ambivalent set in $C[0,1]$. We show that for the prevalent $f\in C[0,1]$ for the generic $y\in f([0,1])$ the level set $f^{-1}(y)$ is perfect. Finally, we answer a question of Darji and White by showing that the set of functions $f \in C[0,1]$ for which there exists a perfect $P_f\subset [0,1]$ such that $f'(x) = \infty$ for all $x \in P_f$ is Haar ambivalent.Comment: 12 page