Let X be a subset of an abelian group and a1, . . . , ah, a ′ 1 , . . . , a h a sequence of 2h elements of X such that a1 + · · ·+ah = a ′ 1 + · · ·+a h . The set X is a Sidon set of order h if, after renumbering, ai = a ′ i for i = 1, . . . , h. For k ≤ h, the set X is a generalized Sidon set of order (h, k), if, after renumbering, ai = a ′ i for i = 1, . . . , k. It is proved that if X is a g...

Let X be a subset of an abelian group and a1, . . . , ah, a ′ 1 , . . . , a h a sequence of 2h elements of X such that a1 + · · ·+ah = a ′ 1 + · · ·+a h . The set X is a Sidon set of order h if, after renumbering, ai = a ′ i for i = 1, . . . , h. For k ≤ h, the set X is a generalized Sidon set of order (h, k), if, after renumbering, ai = a ′ i for i = 1, . . . , k. It is proved that if X is a g...

It is proved that proportions of Sidon sets are I0 subsets of controlled degree. That is, a set E is Sidon if and only if, there are r > 0 and positive integer n such that, for every finite subset F ⊂ E, there is H ⊂ F with the cardinality of H at least r times the cardinality of F and N(H) ≤ n (N(H) is a measure of the degree of being I0). This paper leaves open David Grow’s question of whethe...

A set A of non-negative integers is called a Sidon set if all the sums a1+a2, with a1 ≤ a2 and a1, a2 ∈ A, are distinct. A well-known problem on Sidon sets is the determination of the maximum possible size F (n) of a Sidon subset of [n] = {0, 1, . . . , n− 1}. Results of Chowla, Erdős, Singer and Turán from the 1940s give that F (n) = (1 + o(1)) √ n. We study Sidon subsets of sparse random sets...

A set A of non-negative integers is called a Sidon set if all the sums a1+a2, with a1 ≤ a2 and a1, a2 ∈ A, are distinct. A well-known problem on Sidon sets is the determination of the maximum possible size F (n) of a Sidon subset of [n] = {0, 1, . . . , n− 1}. Results of Chowla, Erdős, Singer and Turán from the 1940s give that F (n) = (1 + o(1)) √ n. We study Sidon subsets of sparse random sets...

Let S be a subset of a group G. We call S a Sidon subset of the first (second) kind, if for any x, y, Z, WE S of which at least 3 are different, xy ~ ZW (xy I "" zwI , resp.). (For abelian groups, the two notions coincide.) If a has a Sidon subset of the second kind with n elements then every n-vertex graph is an induced subgraph of some Cayley graph of G. We prove that a sufficient condition f...

If there is a Sidon subset of the integers Z which has a member of Z as a cluster point in the Bohr compactification of Z, then there is a Sidon subset of Z which is dense in the Bohr compactification. A weaker result holds for quasiindependent and dissociate subsets of Z. It is a long standing open problem whether Sidon subsets of Z can be dense in the Bohr compactification of Z ([LR]). Yitzha...

We classify the compact, connected groups which have infinite central A(p) sets, arithmetically characterize central A(p) sets on certain product groups, and give examples of A(p) sets which are non-Sidon and have unbounded degree. These sets are intimately connected with Figà-Talamanca and Rider's examples of Sidon sets, and stem from the existence of families of tensor product representations...

Let G be any discrete Abelian group. We give several arithmetic characterizations of Sidon sets in G. In particular, we show that a set A is a Sidon set iff there is a number 6 > 0 such that any finite subset A of A contains a subset B Q A with |B| > 6\A\ which is quasiindependent, i.e. such that the only relation of the form ]C\eB e x ^ = '̂ with e\ equal to + 1 or 0, is the trivial one. Let G ...