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Primality testing with Gaussian periods
, 2003
"... The problem of quickly determining whether a given large integer is prime or composite has been of interest for centuries, if not longer. The past 30 years has seen a great deal of progress, leading up to the recent deterministic, polynomialtime algorithm of Agrawal, Kayal, and Saxena [2]. This new ..."
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The problem of quickly determining whether a given large integer is prime or composite has been of interest for centuries, if not longer. The past 30 years has seen a great deal of progress, leading up to the recent deterministic, polynomialtime algorithm of Agrawal, Kayal, and Saxena [2]. This new “AKS test ” for the primality of n involves verifying the
A SUPERADDITIVITY AND SUBMULTIPLICATIVITY PROPERTY FOR CARDINALITIES OF SUMSETS
"... For finite sets of integers A1, A2... An we study the cardinality of the nfold sumset A1 + · · · + An compared to those of n − 1fold sumsets A1 + · · · + Ai−1 + Ai+1 +... An. We prove a superadditivity and a submultiplicativity property for these quantities. We also examine the case when the ..."
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For finite sets of integers A1, A2... An we study the cardinality of the nfold sumset A1 + · · · + An compared to those of n − 1fold sumsets A1 + · · · + Ai−1 + Ai+1 +... An. We prove a superadditivity and a submultiplicativity property for these quantities. We also examine the case when the addition of elements is restricted to an addition graph between the sets.
ON THE POSSIBLE ORDERS OF A BASIS FOR A FINITE CYCLIC GROUP
, 906
"... ABSTRACT. We prove a conjecture of Dukes and Herke concerning the possible orders of a basis for the cyclic group Zn, namely: For each k ∈ N there exists a constant ck> 0 such that, for all n ∈ N, if A ⊆ Zn is a basis of order greater than n/k, then the order of A is within ck of n/l for some int ..."
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ABSTRACT. We prove a conjecture of Dukes and Herke concerning the possible orders of a basis for the cyclic group Zn, namely: For each k ∈ N there exists a constant ck> 0 such that, for all n ∈ N, if A ⊆ Zn is a basis of order greater than n/k, then the order of A is within ck of n/l for some integer l ∈ [1, k]. The proof makes use of various results in additive number theory concerning the growth of sumsets. 1.
SumFree Sets and Related Sets
 Combinatorica
, 1998
"... A set A of integers is sumfree if A"(A+A) = ;. Cameron conjectured that the number of sumfree sets A ` f1; : : : ; Ng is O(2 N=2 ). As a step towards this conjecture, we prove that the number of sets A ` f1; : : : ; Ng satisfying (A +A+A) " (A +A+A+A) = ; is 2 [(N+1)=2] (1 + o(1)). ..."
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A set A of integers is sumfree if A"(A+A) = ;. Cameron conjectured that the number of sumfree sets A ` f1; : : : ; Ng is O(2 N=2 ). As a step towards this conjecture, we prove that the number of sets A ` f1; : : : ; Ng satisfying (A +A+A) " (A +A+A+A) = ; is 2 [(N+1)=2] (1 + o(1)). 1 Introduction We use the notation A+B = fa + b : a 2 A; b 2 Bg; A \Gamma B = fa \Gamma b : a 2 A; b 2 Bg; A + x = fa + x : a 2 Ag; etc., where A; B ` Z and x 2 Z. A symbol "O(: : :)" or "" provided with an index implies a constant depending on the parameter(s) in the index. When there is no index, the implied constant is absolute. A set A ae Z is a sumfree if A " (A + A) = ;: (SF) 1991 Mathematics Subject Classification 11B75 1 Cameron (see [4]) conjectured that the number SF(N) of sumfree sets A ` f1; : : : ; Ng satisfies SF(N) 2 N=2 . Note that the exponent N=2 cannot be improved because any set of odd numbers is sumfree, as well as any subset of f[N=2] + 1; : : : ; Ng. The conjectur...
ON THE NUMBER OF POPULAR DIFFERENCES
"... Abstract. We prove that there exists an absolute constant c> 0 such that for any finite set A ⊆ Z with A  ≥ 2 and any positive integer m < cA / ln A  there are at most m integers b> 0 satisfying (A+b)\A  ≤ m; equivalently, there are at most m positive integers possessing A  −m ( ..."
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Abstract. We prove that there exists an absolute constant c> 0 such that for any finite set A ⊆ Z with A  ≥ 2 and any positive integer m < cA / ln A  there are at most m integers b> 0 satisfying (A+b)\A  ≤ m; equivalently, there are at most m positive integers possessing A  −m (or more) representations as a difference of two elements of A. This is best possible in the sense that for each positive integer m there exists a finite set A ⊆ Z with A > m log2(m/2) such that (A + b) \ A  ≤ m holds for b = 1,...,m+ 1. 1.
The Set of Differences of a Given Set
 AMER. MATH. MONTHLY
, 1999
"... A central problem of combinatorial geometry and additive number theory is to understand the set of sums or differences of a given set of vectors. For example, given a set of m arbitrary vectors A, how big is the set A+A := fa+b : a; b 2 Ag, or the set A\GammaA := fa\Gammab : a; b 2 Ag? By packing ..."
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A central problem of combinatorial geometry and additive number theory is to understand the set of sums or differences of a given set of vectors. For example, given a set of m arbitrary vectors A, how big is the set A+A := fa+b : a; b 2 Ag, or the set A\GammaA := fa\Gammab : a; b 2 Ag? By packing the vectors close together on a lattice one can make these sets small: for example, if A = fa; 2a; 3a; : : : ; (m \Gamma 1)a; mag then A + A and A \Gamma A both have 2m \Gamma 1 elements. On the other hand, if the elements of<F12
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"... Nikolskiitype inequalities for shift invariant function spaces. (English summary) ..."
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Nikolskiitype inequalities for shift invariant function spaces. (English summary)
On the Rate of Decay of the Concentration Function of the Sum of Independent Random Variables
, 2003
"... Abstract. Let X1,..., Xn be i.i.d. integral valued random variables and Sn their sum. In the case when X1 has a moderately large tail of distribution, Deshouillers, Freiman and Yudin gave a uniform upper bound for maxk∈Z Pr{Sn = k} (which can be expressed in term of the Lévy Doeblin concentration o ..."
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Abstract. Let X1,..., Xn be i.i.d. integral valued random variables and Sn their sum. In the case when X1 has a moderately large tail of distribution, Deshouillers, Freiman and Yudin gave a uniform upper bound for maxk∈Z Pr{Sn = k} (which can be expressed in term of the Lévy Doeblin concentration of Sn), under the extra condition that X1 is not essentially supported by an arithmetic progression. The first aim of the paper is to show that this extra condition cannot be simply ruled out. Secondly, it is shown that if X1 has a very large tail (larger than a Cauchytype distribution), then the extra arithmetic condition is not sufficient to guarantee a uniform upper bound for the decay of the concentration of the sum Sn. Proofs are constructive and enhance the connection between additive number theory and probability theory.
MULTIPLE SET ADDITION IN Zp
"... It is shown that there exists an absolute constant H such that for every h>H, every prime p, and every set A ⊆ Zp such that 10 ≤A  ≤p(lnh) 1/2 /(9h 9/4) and hA  ≤ h 3/2 A/(8(lnh) 1/2), the set A is contained in an arithmetic progression modulo p of + 1, where Pj(n) = (j+1)j 2 n − j2 + 1. T ..."
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It is shown that there exists an absolute constant H such that for every h>H, every prime p, and every set A ⊆ Zp such that 10 ≤A  ≤p(lnh) 1/2 /(9h 9/4) and hA  ≤ h 3/2 A/(8(lnh) 1/2), the set A is contained in an arithmetic progression modulo p of + 1, where Pj(n) = (j+1)j 2 n − j2 + 1. This result can be viewed as a generalization of Freiman’s “2.4theorem”. cardinality max1≤j≤h−1 hA−Pj(A) h−j 1.