Wetzel’s sector covers unit arcs

Small Convex Covers for Convex Unit Arcs

More than 40 years ago, Leo Moser stated a geometry problem on the plane asking for the least-area set that contains a congruent copy of every unit arc. The smallest set known, by Norwood and Poole in 2004, has area 0.26043. One interesting related problem is to focus on a set that contains a congruent copy of every convex unit arc. Such a set is called a cover for convex unit arcs. The smalles...

Synthesis by Arcs on the Unit Circle

Take a set of n arcs on the unit circle and consider the characteristic function of the set of n arcs. Think of this as a waveform, taking only the values 0 and 1. Produce a sound by generating a wavetable by sample and hold. You can think of this procedure as a very simple synthesizer. The question is: which sounds can be generated in this way. The surprising answer is: all of them. Thanks to ...

Andrea Calabrese and W . Leo Wetzels

A Connection between Covers of the Integers and Unit Fractions

For integers a and n > 0, let a(n) denote the residue class {x ∈ Z : x ≡ a (mod n)}. Let A be a collection {as(ns)}s=1 of finitely many residue classes such that A covers all the integers at least m times but {as(ns)} s=1 does not. We show that if nk is a period of the covering function wA(x) = |{1 6 s 6 k : x ∈ as(ns)}| then for any r = 0, . . . , nk−1 there are at least m integers in the form...

A Connection between Covers of Z and Unit Fractions

is usually called the covering function of A. Clearly wA(x) is periodic modulo the least common multiple NA of the moduli n1, . . . , nk. As in [S99] we call m(A) = minx∈Z wA(x) the covering multiplicity of A. Let m ∈ Z. If wA(x) > m for all x ∈ Z, then we call A an m-cover of Z. If A is an m-cover of Z but At = {as(ns)}s∈[1,k]\{t} is not (where [a, b] = {x ∈ Z : a 6 x 6 b} for a, b ∈ Z), then ...