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Weakly almost periodic functions and thin sets in discrete groups

1990
*
Transactions of the American Mathematical Society
*

*A*subset E of an infinite discrete group G is called (i) an Rw-set if any bounded function

*on*G supported by E is weakly almost periodic, (ii)

*a*

*weak*p-

*Sidon*set (1 ~ p < 2) if

*on*II (E) the I P -norm ... We show, among other results, that (

*a*) every infinite group G contains an Rw-set which is not an FT-set; (b) countable

*weak*p-

*Sidon*sets, 1 ~ P < 4/3 are F T -sets. ... For abelian G,

*a*

*weak*p-

*Sidon*set is just

*a*p-

*Sidon*set as defined by Edwards and Ross [7] .

*Note*also that (

*weak*) I-

*Sidon*sets are just (

*weak*)

*Sidon*sets as defined by Picardello [15] . ...

##
###
Weakly Almost Periodic Functions and Thin Sets in Discrete Groups

1990
*
Transactions of the American Mathematical Society
*

*A*subset E of an infinite discrete group G is called (i) an Rw-set if any bounded function

*on*G supported by E is weakly almost periodic, (ii)

*a*

*weak*p-

*Sidon*set (1 ~ p < 2) if

*on*II (E) the I P -norm ... We show, among other results, that (

*a*) every infinite group G contains an Rw-set which is not an FT-set; (b) countable

*weak*p-

*Sidon*sets, 1 ~ P < 4/3 are F T -sets. ... For abelian G,

*a*

*weak*p-

*Sidon*set is just

*a*p-

*Sidon*set as defined by Edwards and Ross [7] .

*Note*also that (

*weak*) I-

*Sidon*sets are just (

*weak*)

*Sidon*sets as defined by Picardello [15] . ...

##
###
On Fourier-Stieltjes transforms of continuous measures

1977
*
Journal of Functional Analysis
*

The continuous measure whose transform equals 1

doi:10.1016/0022-1236(77)90023-4
fatcat:wg6orvzxinbqzpmncdps4rtp54
*on**a*non-*Sidon*portion of (JL, Ej C E will be obtained as*a**weak** limit of trigonometric polynomials that are inductively defined below. ... Fix*a**sequence*of positive prime integers, (pj), tending monotonically to infinity, and let EC E be*a*non-*Sidon*set. ... The continuous measure whose transform equals 1*on**a*non-*Sidon*portion of (JL, Ej C E will be obtained as*a**weak** limit of trigonometric polynomials that are inductively defined below. ...##
###
Page 1784 of Mathematical Reviews Vol. 55, Issue 6
[page]

1978
*
Mathematical Reviews
*

Then the quantity o(£)= sup{inf{||z||:

*A*=*on*E}: pe 1 °(E), |\—|| . <1} is finite.*A*lacunary*sequence*in the integer group is an example of*a**Sidon*set. ... Lecture*Notes*in Pure and Appl. Math., Vol. 2, Dekker, New York, 1971. Kronecker, Helson, Dirichlet,*weak*Kronecker, and*weak*Dirichlet sets are considered. ...##
###
Largest cliques in connected supermagic graphs

2005
*
Discrete Mathematics & Theoretical Computer Science
*

Bounds

doi:10.46298/dmtcs.3414
fatcat:z45d6kkqobakpps5uokiuslxam
*on**Sidon*sets show that the order of such*a*graph is at least $n^2+o(n^2)$. ... Moreover it can be required that the graph admits*a*supermagic labelling $f$, which satisfies the additional condition $f(V)=[1,|V|]$. ...*On*the other hand, the known constructions of dense*Sidon*sets together with results*on*the distribution of primes provide in particular*weak**Sidon*sets of cardinality n in [1, N ] with N = n 2 + o(n ...##
###
Arens regularity and weak sequential completeness for quotients of the Fourier algebra

2000
*
Illinois Journal of Mathematics
*

This is

doi:10.1215/ijm/1255984689
fatcat:jlxpdd5a3jhf5o4vajvp3ltpl4
*a*study of Arens regularity in the context of quotients of the Fourier algebra*on**a*non-discrete locally compact abelian group (or compact group). (1) If*a*compact set E of G is of bounded synthesis ... and is the support of*a*pseudofunction, then*A*(E) is weakly sequentially complete. ( This implies that every point of E is*a*Day point.) (2) If*a*compact set E supports*a*synthesizable pseudofunction ... (ii) Any*weak** (tr(*A*(G)*,*A*(G)**)) accumulation point of*a*1-tenting*sequence**on**A*(G) is an invariant mean*on**A*(G)* [12] , [18]. See also Remark 2.5.1. ...##
###
The Ubiquity of Sidon Sets That Are Not I_0
[article]

2016
*
arXiv
*
pre-print

We prove that every infinite, discrete abelian group admits

arXiv:1602.04241v1
fatcat:khzmr6hc5zhfjg252maubz57dy
*a*pair of I_0 sets whose union is not I_0. In particular, this implies that every such group contains*a**Sidon*set that is not I_0. ... We claim that {Π(λ n + χ n )} is*weak*ε q -Kronecker. To prove this,*note*that*a*character g*on*Λ is specified as g = {g n }, with each g n*a*character*on*Ω β n . ... Introduction*A*subset E of*a*discrete abelian group Γ with compact dual group G is said to be*a**Sidon*set if every bounded E-*sequence*can be interpolated by the Fourier transform of*a*measure*on*G. ...##
###
Weakly almost periodic functions and Fourier-Stieltjes algebras of locally compact groups

1982
*
Transactions of the American Mathematical Society
*

But we do not have

doi:10.1090/s0002-9947-1982-0670924-2
fatcat:n6hafhgi75fs5k7q2vxs3oiwv4
*a*proof yet. ... Therefore, the above proposition states that if 5 contains large squares then it is not*a**weak**Sidon*set. When G is abelian,*a**weak**Sidon*set is just*a**Sidon*set in the usual sense. ... Using standard results in functional analysis,*one*sees that*a*set S in G is*weak**Sidon*if and only if II • II, and II • ||" are equivalent*on*lx(S). ...##
###
Largest cliques in connected supermagic graphs

2007
*
European journal of combinatorics (Print)
*

Bounds

doi:10.1016/j.ejc.2007.04.006
fatcat:efpa6zb6zzbgnijap22k6xlufi
*on**Sidon*sets show that the order of such*a*graph is at least n 2 +o(n 2 ). ...*A*graph G = (V, E) is said to be magic if there exists an integer labeling f : Enomoto, Masuda and Nakamigawa proved that there are magic graphs of order at most 3n 2 + o(n 2 ) which contain*a*complete ... Let*A*= {*a*1 <*a*2 < • • • <*a*n } ⊂ [1, N ] be*a**weak**Sidon*set. Since*A*−*a*1 + 1 is*a**weak**Sidon*set as well, we may assume that*a*1 = 1. ...##
###
Completely Sidon sets in C^*-algebras (New title)
[article]

2018
*
arXiv
*
pre-print

*A*

*sequence*in

*a*C^*-algebra

*A*is called completely

*Sidon*if its span in

*A*is completely isomorphic to the operator space version of the space ℓ_1 (i.e. ℓ_1 equipped with its maximal operator space structure ... Our main result is

*a*generalization to this context of Drury's classical theorem stating that

*Sidon*sets are stable under finite unions. ... The predual M 1 * is the subset of M 1 * formed of the

*weak** continuous functionals

*on*M 1 . ...

##
###
On uniformly bounded orthonormal Sidon systems
[article]

2016
*
arXiv
*
pre-print

This sharpens their result that it is 5-fold tensor

arXiv:1602.02430v6
fatcat:5c2p552hgrc5hdazuc7m5cuivq
*Sidon*. The proof is somewhat reminiscent of the author's original*one*for (Abelian) group characters, based*on*ideas due to Drury and Rider. ... In the latter setting we also include*a*new proof of Rider's unpublished result that randomly*Sidon*sets are*Sidon*, which implies that the union of two*Sidon*sets is*Sidon*. ... Let (ψ 1 n ) be*a**Sidon**sequence*. Then for any*sequence*(ψ 2 n ) such that δ = inf ψ 2 n 1 > 0, the*sequence*(ψ 1 n ⊗ ψ 2 n ) is*Sidon*. ...##
###
On uniformly bounded orthonormal Sidon systems

2017
*
Mathematical Research Letters
*

We also show that

doi:10.4310/mrl.2017.v24.n3.a13
fatcat:r45f2j5quzbz7f6k6ae4xon5bu
*a*uniformly bounded orthonormal system is randomly*Sidon*i it is ⊗ 4 tensor*Sidon*, or equivalently ⊗ k -*Sidon*for some (or all) k ≥ 4. ... In the latter setting we also include*a*new proof of Rider's unpublished result that randomly*Sidon*sets are*Sidon*, which implies that the union of two*Sidon*sets is*Sidon*. ...*A**sequence*(ϕ n ) of distinct irreducible representations*on**a*compact group is*Sidon*iff it is randomly*Sidon*. Corollary 5. e ij ⊗ e ij . ...##
###
Sidon Sequences and Doubly Periodic Two-Dimensional Synchronization Patterns
[article]

2011
*
arXiv
*
pre-print

*A*few constructions of two-dimensional synchronization patterns are based

*on*these

*sequences*. ... We also present

*a*new construction for

*Sidon*

*sequences*over an alphabet of size q(q-1), where q is

*a*power of

*a*prime. ... D is

*a*

*Sidon*

*sequence*(or

*a*B 2 -

*sequence*) over

*A*if all the sums

*a*i1 +

*a*i2 with 1 ≤ i 1 ≤ i 2 ≤ m are distinct (if i 1 < i 2 in the definition the

*sequence*is called

*a*

*weak*

*Sidon*

*sequences*). ...

##
###
An upper bound on the size of Sidon sets
[article]

2021
*
arXiv
*
pre-print

We show that the maximum size of

arXiv:2103.15850v2
fatcat:pnfncdgwejarldzcrspmz7uuv4
*a**Sidon*set of { 1, 2, ..., n} is at most √(n)+ 0.998n^1/4 for sufficiently large n. ... In this entry point into the subject, combining two elementary proofs, we decrease the gap between the upper and lower bounds by 0.2% in*a*classical combinatorial number theory problem. ...*Note*that for*Sidon*sets we need this to hold for i ≤ j. Denote by W (n) the size of the largest*weak**Sidon*subset of [n]. ...##
###
Fourier-Stieltjes Transforms which vanish at infinity off certain sets

1978
*
Glasgow Mathematical Journal
*

The method of Theorem 2 shows that the union of

doi:10.1017/s0017089500003360
fatcat:gwthzt6htfextdnunouwp7oq4u
*a**weak*Rajchman set and*a**Sidon*set is*a**weak*Rajchman set. ...*A*proof for arbitrary discrete T that*a**Sidon*set is*a*Rajchman set can be based*on*Theorem 1.4 of [16, p. 8] and the method of proof of the present example. strong Riesz set. ...*Note*added in proof. The author has recently learned of the work of Keiji Izuchi,*Sidon*sets and small M 0 -sets, Sci. Rep. Tokyo Kyoiku Daigaku Sect.*A*12 (1974), 146-148. ...
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