Difference bases in cyclic groups

Abstract

A subset B of a group G is called a difference basis of G if each element g ∈ G can be written as the difference g = ab −1 of some elements a, b ∈ B. The smallest cardinality |B| of a difference basis B ⊂ G is called the difference size of G and is denoted by ∆ [G]. The fraction ð[G] := ∆[G]/ |G| is called the difference characteristic of G. We prove that for every n ∈ N the dihedral group D 2n of order 2n has the difference characteristic√ 586 ≈ 1.983. Moreover, if n ≥ 2 · 10 15 , then ð[D 2n ] < 4 √ 6 ≈ 1.633. Also we calculate the difference sizes and characteristics of all dihedral groups of cardinality ≤ 80. 1991 Mathematics Subject Classification. 05B10, 05E15, 20D60.

On symmetric intersecting families

“…The constructions of Singer [25] and Bose [3] yield affirmative answers to Question 4.4 when n is of the form q 2 + q + 1 or q 2 − 1 respectively, where q is a prime power, and a construction due to Ruzsa [23] does so when n is of the form p 2 − p, where p is prime; as observed by Banakh and Gavrylkiv [1], these constructions of Singer, Bose and Ruzsa yield efficient difference covers as well, so we also have affirmative answers to Questions 4.3 and 4.1 for all n of the aforementioned form.…”

“…In view of Lemma 4.2, we are led to the following question, which being a natural question in its own right, has also occurred independently to others; see [1], for instance. √ n for all n ∈ N?…”

“…We saw from the projective construction earlier that if n = q 2 + q + 1 for some prime power q, then s(n, k) > 0 if and only if k ≥ q + 1; in particular, this implies that g(n) = (1 + o(1)) √ n. We also know from the observation of Banakh and Gavrylkiv [1] mentioned earlier that g(n) = (1 + o(1)) √ n if n = q 2 − 1 for some prime power q, or if n = p 2 − p for some prime p. Consequently, we have an affirmative answer to Question 4.1 for any n of the form q 2 + q + 1 or q 2 − 1 for some prime power q, or of the form p 2 − p for some prime p.…”

Difference bases in finite Abelian groups

“…Table 1. Difference sizes of groups of order ≤ 13 G: In [2] the difference sizes of the order-intervals [1, n] ∩ Z were applied to give upper bounds for the difference sizes of finite cyclic groups. Proposition 2.4.…”

“…The following theorem was derived in [2] from the classical results of Singer [13], Bose, Chowla [4], [5] and Rusza [12].…”