235 research outputs found
The number of multiplicative Sidon sets of integers
A set of natural numbers is multiplicative Sidon if the products of all
pairs in are distinct. Erd\H{o}s in 1938 studied the maximum size of a
multiplicative Sidon subset of , which was later determined up
to the lower order term: . We
show that the number of multiplicative Sidon subsets of is
for a certain function
which we specify. This is a rare example in which
the order of magnitude of the lower order term in the exponent is determined.
It resolves the enumeration problem for multiplicative Sidon sets initiated by
Cameron and Erd\H{o}s in the 80s.
We also investigate its extension for generalised multiplicative Sidon sets.
Denote by , , the number of multiplicative -Sidon subsets of
. We show that for some
we define explicitly. Our proof is elementary.Comment: 20 page
On modular k-free sets
Let and be integers. A set is
-free if for all in , . We determine the maximal
cardinality of such a set when and are coprime. We also study several
particular cases and we propose an efficient algorithm for solving the general
case. We finally give the asymptotic behaviour of the minimal size of a
-free set in which is maximal for inclusion
Constructions of Generalized Sidon Sets
We give explicit constructions of sets S with the property that for each
integer k, there are at most g solutions to k=s_1+s_2, s_i\in S; such sets are
called Sidon sets if g=2 and generalized Sidon sets if g\ge 3. We extend to
generalized Sidon sets the Sidon-set constructions of Singer, Bose, and Ruzsa.
We also further optimize Koulantzakis' idea of interleaving several copies of a
Sidon set, extending the improvements of Cilleruelo & Ruzsa & Trujillo, Jia,
and Habsieger & Plagne. The resulting constructions yield the largest known
generalized Sidon sets in virtually all cases.Comment: 15 pages, 1 figure (revision fixes typos, adds a few details, and
adjusts notation
On Multiplicative Sidon Sets
Fix integers with . A set is
\emph{-multiplicative} if for all . For all ,
we determine an -multiplicative set with maximum cardinality in ,
and conclude that the maximum density of an -multiplicative set is
. For , a set
is \emph{-multiplicative} if implies and for
all and , and . For and
coprime, we give an O(1) time algorithm to approximate the maximum density of
an -multiplicative set to arbitrary given precision
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