When almost all sets are difference dominated

When almost all sets are difference dominated

We investigate the relationship between the sizes of the sum and difference sets attached to a subset of {0, 1, ..., N}, chosen randomly according to a binomial model with parameter p(N), with N = o(p(N)). We show that the random subset is almost surely difference dominated, as N → ∞, for any choice of p(N) tending to zero, thus confirming a conjecture of Martin and O’Bryant. Furthermore, we ex...

When Generalized Sumsets Are Difference Dominated

We study the relationship between the number of minus signs in a generalized sumset, A+ · · ·+A− · · · −A, and its cardinality; without loss of generality we may assume there are at least as many positive signs as negative signs. As addition is commutative and subtraction is not, we expect that for most A a combination with more minus signs has more elements than one with fewer; however, recent...

Codebooks from almost difference sets

In direct spread CDMA systems, codebooks meeting the Welch bounds are used to distinguish among the signals of different users. Recently, constructions of codebooks with difference sets meeting Welch’s bound on the maximum cross-correlation amplitude were developed. In this paper, a generic construction of codebooks using almost difference sets is considered and several classes of codebooks nea...

Genetically-designed Arbitrary Length Almost Difference Sets

Almost All Palindromes Are Composite

We study the distribution of palindromic numbers (with respect to a fixed base g ≥ 2) over certain congruence classes, and we derive a nontrivial upper bound for the number of prime palindromes n ≤ x as x → ∞. Our results show that almost all palindromes in a given base are composite. ∗MSC Numbers: 11A63, 11L07, 11N69 †Corresponding author 1