Functions not Constant on Fractal Quasi-Arcs of Critical Points

Distribution of Points on Arcs

Let z1, . . . , zN be complex numbers situated on the unit circle |z| = 1, and write S := z1 + · · · + zN . Generalizing a well-known lemma by Freiman, we prove the following. (i) Suppose that any open arc of length φ ∈ (0, π] of the unit circle contains at most n of the numbers z1, . . . , zN . Then |S| ≤ 2n−N + 2(N − n) cos(φ/2). (ii) Suppose that any open arc of length π of the unit circle c...

On the Critical Points of Harmonic Functions

Critical Points of Functions on Singular Spaces

We compare and contrast various notions of the “critical locus” of a complex analytic function on a singular space. After choosing a topological variant as our primary notion of the critical locus, we justify our choice by generalizing Lê and Saito’s result that constant Milnor number implies that Thom’s af condition is satisfied.

Critical points of simple functions

Existence of Quasi-arcs

We show that doubling, linearly connected metric spaces are quasiarc connected. This gives a new and short proof of a theorem of Tukia.