Lusin type theorems for Radon measures

We add to the literature the following observation. If $\mu$ is a singular measure on $\mathbb{R}^n$ which assigns measure zero to every porous set and $f:\mathbb{R}^n\rightarrow\mathbb{R}$ is a Lipschitz function which is non-differentiable $\mu$-a.e. then for every $C^1$ function $g:\mathbb{R}^n\rightarrow\mathbb{R}$ it holds $$\mu\{x\in\mathbb{R}^n: f(x)=g(x)\}=0.$$ In other words the Lusin type approximation property of Lipschitz functions with $C^1$ functions does not hold with respect to a general Radon measure

Wandering domains for composition of entire functions

C. Bishop has constructed an example of an entire function f in Eremenko-Lyubich class with at least two grand orbits of oscillating wandering domains. In this paper we show that his example has exactly two such orbits, that is, f has no unexpected wandering domains. We apply this result to the classical problem of relating the Julia sets of composite functions with the Julia set of its members. More precisely, we show the existence of two entire maps f and g in Eremenko-Lyubich class such that the Fatou set of f compose with g has a wandering domain, while all Fatou components of f or g are preperiodic. This complements a result of A. Singh and results of W. Bergweiler and A. Hinkkanen related to this problem.Comment: 21 pages, 3 figure

On the difference between permutation polynomials over finite fields

The well-known Chowla and Zassenhaus conjecture, proven by Cohen in 1990, states that if $p>(d^2-3d+4)^2$, then there is no complete mapping polynomial $f$ in \Fp[x] of degree $d\ge 2$. For arbitrary finite fields \Fq, a similar non-existence result is obtained recently by I\c s\i k, Topuzo\u glu and Winterhof in terms of the Carlitz rank of $f$. Cohen, Mullen and Shiue generalized the Chowla-Zassenhaus-Cohen Theorem significantly in 1995, by considering differences of permutation polynomials. More precisely, they showed that if $f$ and $f+g$ are both permutation polynomials of degree $d\ge 2$ over \Fp, with $p>(d^2-3d+4)^2$, then the degree $k$ of $g$ satisfies $k \geq 3d/5$, unless $g$ is constant. In this article, assuming $f$ and $f+g$ are permutation polynomials in \Fq[x], we give lower bounds for $k$ in terms of the Carlitz rank of $f$ and $q$. Our results generalize the above mentioned result of I\c s\i k et al. We also show for a special class of polynomials $f$ of Carlitz rank $n \geq 1$ that if $f+x^k$ is a permutation of \Fq, with $\gcd(k+1, q-1)=1$, then $k\geq (q-n)/(n+3)$

Bounding Helly numbers via Betti numbers

We show that very weak topological assumptions are enough to ensure the existence of a Helly-type theorem. More precisely, we show that for any non-negative integers $b$ and $d$ there exists an integer $h(b,d)$ such that the following holds. If $\mathcal F$ is a finite family of subsets of $\mathbb R^d$ such that $\tilde\beta_i\left(\bigcap\mathcal G\right) \le b$ for any $\mathcal G \subsetneq \mathcal F$ and every $0 \le i \le \lceil d/2 \rceil-1$ then $\mathcal F$ has Helly number at most $h(b,d)$. Here $\tilde\beta_i$ denotes the reduced $\mathbb Z_2$-Betti numbers (with singular homology). These topological conditions are sharp: not controlling any of these $\lceil d/2 \rceil$ first Betti numbers allow for families with unbounded Helly number. Our proofs combine homological non-embeddability results with a Ramsey-based approach to build, given an arbitrary simplicial complex $K$, some well-behaved chain map $C_*(K) \to C_*(\mathbb R^d)$.Comment: 29 pages, 8 figure