On Keller's conjecture for certain cyclic groups

Abstract

Keller (6) considered a generalisation of a problem of Minkowski (7) concerning the filling of Rn by congruent cubes. Hajós (4) reduced Minkowski's conjecture to a problem concerning the factorization of finite abelian groups and then solved this problem. In a similar manner Hajós (5) reduced Keller's conjecture to a problem in the factorization of finite abelian groups, but this problem remains unsolved, in general. It occurs also as Problem 80 in Fuchs (3). Seitz (10) has obtained a solution for cyclic groups of prime power order. In this paper we present a solution for cyclic groups whose order is the product of two prime powers.

Universal Spectra and Tijdeman's Conjecture on Factorization of Cyclic Groups

“…[18] for references. As indicated in the introduction, this conjecture was actually made earlier by Sands [16], who proved it also holds in the cases m = p n q k (n, k ≥ 1).…”

“…Recently, Coven and Meyerowitz [3] observed that a conjecture equivalent to Tijdeman's Conjecture (mod m) had been made earlier, by Sands [16] in 1977. Sands proved this conjecture holds for all m divisible by at most two distinct primes.…”

The Favard length of product Cantor sets

“…[2], [5], [6], [9], [11], [12], [14]. In this paper, we will rely on (5.2) to provide a key estimate in the next subsection.…”

“…Multiplying both sides of (5.7) by 1 − x 1/M K m , we get 11) where Q(u) = 1 + (u − 1)P (u) is a polynomial of degree at most r + q. The left hand side of (5.11) is bounded in absolute value by…”

ELEMENTARY p-GROUPS WITH THE RÉDEI PROPERTY

“…Sands [11] showed that if p and q are distinct primes then groups of type (p α , q β ) have the Rédei property. Szabó [12] proved that certain cyclic groups do not have the Rédei property, and later characterized all finite cyclic groups with the Rédei property [13].…”