Random Sidon Sequences

Godbole, Anant P.; Janson, Svante; Locantore, Nicholas; Rapoport, Rebecca

doi:10.1006/jnth.1998.2325

Cited by 17 publications

(33 citation statements)

References 9 publications

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“…We note that the value of threshold probability in Theorem 1.6 fits with the results in [6] for Sidon sets (i.e., k = 1). This is in contrast with the results obtained in Theorems 1.3 and 1.4.…”

Section: Introduction and Resultssupporting

confidence: 71%

“…For k = 1 the above definition corresponds to the binomial model of random subsets. Godbole, Janson, Locantore and Rapoport [6] showed, among more general results, that N −3/4 is the threshold probability for a random set in [N ] to be a Sidon set. In the case of random systems we obtain the threshold probability for all k ≥ 2.…”

Section: Introduction and Resultsmentioning

confidence: 99%

“…Especially in light of the results obtained by Godbole et al in [6], the initial assumption might be that this corresponds to the case 2h for general h, which would lead to a threshold at…”

Section: Results For H-fold Sumsetsmentioning

confidence: 99%

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Sidon set systems

2020

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A family A of k-subsets of {1, 2, . . . , N } is a Sidon system if the sumsets A+ B, A, B ∈ A are pairwise distinct. We show that the largest cardinality F k (N ) of a Sidon system of k-. More precise bounds on F k (N ) are obtained for k ≤ 3. We also obtain the threshold probability for a random system to be Sidon for k ≥ 2.This upper bound is tight for k = 2, that is, F 2 (N ) = 2N − 3. We believe that it is asymptotically sharp for any k ≥ 2, but we are only able to prove this for k = 2 and k = 3. For k ≥ 4, we prove that N k−1 is the right order of magnitude.Mathematics Subject Classification (2010): Primary 11B13; Secondary 05B10.

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Section: Introduction and Resultssupporting

confidence: 71%

Section: Introduction and Resultsmentioning

confidence: 99%

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Sidon set systems

2020

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“…, the results of [2] imply that all pairs (x, y) in a random set a.s. generate different values f (x, y), for any f , so that |f (A)| = |g(A)| a.s. for any f and g.…”

Section: Introductionmentioning

confidence: 90%

When almost all sets are difference dominated

Hegarty

Miller

2009

Random Struct Algorithms

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ABSTRACT: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 −1 = 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. The proofs use recent strong concentration results.Furthermore, we exhibit a threshold phenomenon regarding the ratio of the size of the difference to the sumset. If p(N) = o(N −1/2 ) then almost all sums and differences in the random subset are almost surely distinct and, in particular, the difference set is almost surely about twice as large as the sumset. If N)) then both the sum and difference sets almost surely have size (2N +1)−O(p(N) −2 ), and so the ratio in question is almost surely very close to one. If p(N) = c · N −1/2 then as c increases from zero to infinity (i.e., as the threshold is crossed), the same ratio almost surely decreases continuously from two to one according to an explicitly given function of c.We also extend our results to the comparison of the generalized difference sets attached to an arbitrary pair of binary linear forms. For certain pairs of forms f and g, we show that there in fact exists a sharp threshold at c f ,g · N −1/2 , for some computable constant c f ,g , such that one form almost surely dominates below the threshold and the other almost surely above it.The heart of our approach involves using different tools to obtain strong concentration of the sizes of the sum and difference sets about their mean values, for various ranges of the parameter p.

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“…Godbole, Janson, Locantore and Rapoport studied the threshold function for the Sidon property and gave the exact probability distribution in 1999 (see [2]): Clearly, a subset H ⊂ [1, n] is a degenerate 2-cube iff it is an AP 3 . Moreover, an easy argument gives that the threshold function for the event "AP 3 -free" is p n = n −2/3 .…”

Section: Introductionmentioning

confidence: 99%

Non-degenerate Hilbert cubes in random sets

Sándor¹

2007

J. Théor. Nombres Bordeaux

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Random Sidon Sequences

Cited by 17 publications

References 9 publications

Sidon set systems

Sidon set systems

When almost all sets are difference dominated

Non-degenerate Hilbert cubes in random sets

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