Sidon set

We investigate on the long-standing conjecture of Paul Erdős concerning distinct subset sums. For any set A = {a 1 < • • • < an} ⊂ N with all subset sums distinct, we prove the lower bound max(A) ≥ c • 2 n for an absolute constant c > 0, confirming Erdős's conjecture. Our approach introduces a novel application of the circle method to additive combinatorics, establishing a fundamental connection between the distribution of subset sums and exponential sums over major arcs.

Let G be a compact group and G its dual (we will use our notation from [3]). For a e G, let T a e a. Then T~ is a continuous homomorphism of G into U(G), the group of G • n~ unitary matrices. We use T~(x)q to denote the matrix entries of T,(x), 1 < i,j < G. Let ga(x) = Tr(Ta(x))

A set A = A k,n ⊂ [n] ∪ {0} is said to be an additive h-basis if each element in {0, 1,. .. , hn} can be written as an h-sum of elements of A in at least one way. We seek multiple representations as h-sums, and, in this paper we make a start by restricting ourselves to h = 2. We say that A is said to be a truncated (α, 2, g) additive basis if each j ∈ [αn, (2 − α)n] can be represented as a 2-sum of elements of A k,n in at least g ways. In this paper, we provide sharp asymptotics for the event that a randomly selected set is a truncated (α, 2, g) additive basis with high or low probability.

A set A ⊆ [n] ∪ {0} is said to be a 2-additive basis for [n] if each j ∈ [n] can be written as j = x + y, x, y ∈ A, x ≤ y. If we pick each integer in [n] ∪ {0} independently with probability p = p n → 0, thus getting a random set A, what is the probability that we have obtained a 2-additive basis? We address this question when the target sum-set is [(1 − α)n, (1 + α)n] (or equivalently [αn, (2 − α)n]) for some 0 < α < 1. Under either model, the Stein-Chen method of Poisson approximation is used, in conjunction with Janson's inequalities, to tease out a very sharp threshold for the emergence of a 2-additive basis. Generalizations to k-additive bases are then given.

A set A ⊆ [n] ∪ {0} is said to be a 2-additive basis for [n] if each j ∈ [n] can be written as j = x + y, x, y ∈ A, x ≤ y. If we pick each integer in [n] ∪ {0} independently with probability p = p n → 0, thus getting a random set A, what is the probability that we have obtained a 2-additive basis? We address this question when the target sum-set is [(1 − α)n, (1 + α)n] (or equivalently [αn, (2 − α)n]) for some 0 < α < 1. Under either model, the Stein-Chen method of Poisson approximation is used, in conjunction with Janson's inequalities, to tease out a very sharp threshold for the emergence of a 2-additive basis. Generalizations to k-additive bases are then given.

A set A ⊆ [n] ∪ {0} is said to be a 2-additive basis for [n] if each j ∈ [n] can be written as j = x + y, x, y ∈ A, x ≤ y. If we pick each integer in [n] ∪ {0} independently with probability p = p n → 0, thus getting a random set A, what is the probability that we have obtained a 2-additive basis? We address this question when the target sum-set is [(1 − α)n, (1 + α)n] (or equivalently [αn, (2 − α)n]) for some 0 < α < 1. Under either model, the Stein-Chen method of Poisson approximation is used, in conjunction with Janson's inequalities, to tease out a very sharp threshold for the emergence of a 2-additive basis. Generalizations to k-additive bases are then given.

We consider anti-Ramsey type problems for k-uniform hypergraphs. A subset Y of an n-element set X is totally multicolored, if the restriction of a coloring of k-element subsets of X to [Y ] k is a one-to-one coloring. We study the maximum size of totally multicolored subsets for colorings which satisfy the following restriction: any two distinct k-element subsets with at least h elements in the intersection are colored differently. It is shown that the size of the largest totally multicolored subset can be bounded from below by c k } (ln n) 1Â(2k&1) } n (k&h)Â(2k&1) , where c k is a positive constant and this lower bound is sharp (up to a multiplicative constant). Extending earlier results of present and other authors, various related problems concerning edge-colorings of complete graphs are considered. We consider also some application to Sidon sets in arbitrary groups. We show for example the existence of a proper coloring of [X ] 2 with no totally multicolored subset of size 2.21n 1Â3 (ln n) 1Â3 , provided n n 0. This sharpens an earlier result of L. Babai and is up to a multiplicative constant the best possible (this follows e.g. from the above mentioned result applied with k=2 and h=1). As a tool for proving the existence of such a coloring we derive an upper bound for permanents of rectangular (0, 1)-matrices

A system of r-element subsets (blocks) of an n-element set X n is called a Tura n (n, k, r)-system if every k-element subset of X n contains at least one of the blocks. The Tura n number T(n, k, r) is the minimum size of such a system. We prove upper estimates: T(n, r+1, r) (1+o(1)

A Sidon set is a subset of an Abelian group with the property that each sum of two distinct elements is distinct. We construct a small maximal Sidon set of size O((n • 2 n) 1/3) in the group Z n 2 , generalizing a result of Ruzsa concerning maximal Sidon sets in the integers.

Let G be a compact group and feL 2 (G). We prove that given p < co there exists a unitary transformation U of L\G) into L\G), which commutes with left translations and such that UfeL p. The proof is based on techniques developed by S. Helgason for a similar question. The result stated above, which is an extension of a theorem of Littlewood for the unit circle is then applied to the study of lacunary Fourier series.

For each positive even integer j there is an infinite arithmetic sequence of dimensions d for which we construct a j-simplex of maximum volume in the d-dimensional unit cube. For fixed d, all of these maximal j-simplices have the same Gram matrix, which is a multiple of I+J. For j even, a new upper bound for the volume of a j-simplex in the d-dimensional unit cube is given.

We establish a general and optimal lower bound for the complete sum of the probabilities of k-intersections of n events. We then describe various applications to additive and multiplicative number theory, graph theory, coding theory, study of lattice points on circles, and divisors of polynomials

Under consideration is the problem of constructing a square Boolean matrix A of order n without "rectangles" (it is a matrix whose every submatrix of the elements that are in any two rows and two columns does not consist of 1s). A linear transformation modulo two defined by A has complexity o(ν(A) − n) in the base {⊕}, where ν(A) is the weight of A, i.e., the number of 1s (the matrices without rectangles are called thin). Two constructions for solving this problem are given. In the first construction, n = p 2 where p is an odd prime. The corresponding matrix H p has weight p 3 and generates the linear transformation of complexity O(p 2 log p log log p). In the second construction, the matrix has weight nk where k is the cardinality of a Sidon set in Z n. We may assume that k = Θ(√ n); there are examples of the sets of cardinality k ∼ √ n for some n. The corresponding linear transformation has complexity O(n log n log log n). Some generalizations of this problem are considered.

Let G be a compact group and let ^PçG. We consider the inequalities 1 < k(P) < (S.aep 4,2)1/2> where k(P) denotes the Sidon constant of P. The condition k(P) = 1 essentially characterizes an example of Figà-Talamanca and Rider. The condition k(P) = (2o6i. d?)1/2 for finite P is equivalent to the existence of certain interesting functions on G. We show that k(G)-|6*|1''2 for a very large class of finite groups G, and this implies the existence of "G-circulant" unitary matrices whose entries all have modulus |G|~'/2.

We consider anti-Ramsey type problems for k-uniform hypergraphs. A subset Y of an n-element set X is totally multicolored, if the restriction of a coloring of k-element subsets of X to [Y ] k is a one-to-one coloring. We study the maximum size of totally multicolored subsets for colorings which satisfy the following restriction: any two distinct k-element subsets with at least h elements in the intersection are colored differently. It is shown that the size of the largest totally multicolored subset can be bounded from below by where c k is a positive constant and this lower bound is sharp (up to a multiplicative constant). Extending earlier results of present and other authors, various related problems concerning edge-colorings of complete graphs are considered. We consider also some application to Sidon sets in arbitrary groups. We show for example the existence of a proper coloring of [X ] 2 with no totally multicolored subset of size 2.21n 1Â3 (ln n) 1Â3 , provided n n 0 . This sharpens an earlier result of L. Babai and is up to a multiplicative constant the best possible (this follows e.g. from the above mentioned result applied with k=2 and h=1). As a tool for proving the existence of such a coloring we derive an upper bound for permanents of rectangular (0, 1)-matrices 1996 Academic Press, Inc. A typical anti-Ramsey type question, cf. [ESS75], [Ra75], is the following: Given is a graph or a hypergraph, where the edges are colored with certain restrictions on the coloring like, for example, edges of the same article no. 0049 209 0097-3165Â96 18.00

A set $\mathcal{A}$ is said to be an additive $h$-basis if each element in $\{0,1,\ldots,hn\}$ can be written as an $h$-sum of elements of $\mathcal{A}$ in {\it at least} one way. We seek multiple representations as $h$-sums, and, in this paper we make a start by restricting ourselves to $h=2$. We say that $\mathcal{A}$ is said to be a truncated $(\alpha,2,g)$ additive basis if each $j\in[\alpha n, (2-\alpha)n]$ can be represented as a $2$-sum of elements of $\mathcal{A}$ in at least $g$ ways. In this paper, we provide sharp asymptotics for the event that a randomly selected set is a truncated $(\alpha,2,g)$ additive basis with high or low probability.

A set A=A_{k,n} in [n]\cup{0} is said to be an additive k-basis if each element in {0,1,...,kn} can be written as a k-sum of elements of A in at least one way. Seeking multiple representations as k-sums, and given any function phi(n), with lim(phi(n))=infinity, we say that A is a truncated phi(n)-representative k-basis for [n] if for each j in [alpha n, (k-alpha)n] the number of ways that j can be represented as a k-sum of elements of A_{k,n} is Theta(phi(n)). In this paper, we follow tradition and focus on the case phi(n)=log n, and show that a randomly selected set in an appropriate probability space is a truncated log-representative basis with probability that tends to one as n tends to infinity. This result is a finite version of a result proved by Erdos (1956) and extended by Erdos and Tetali (1990).

A set A ⊆ [n] ∪ {0} is said to be a 2-additive basis for [n] if each j ∈ [n] can be written as j = x + y, x, y ∈ A, x ≤ y. If we pick each integer in [n] ∪ {0} independently with probability p = p n → 0, thus getting a random set A, what is the probability that we have obtained a 2-additive basis? We address this question when the target sum-set is [(1 − α)n, (1 + α)n] (or equivalently [αn, (2 − α)n]) for some 0 < α < 1. Under either model, the Stein-Chen method of Poisson approximation is used, in conjunction with Janson's inequalities, to tease out a very sharp threshold for the emergence of a 2-additive basis. Generalizations to k-additive bases are then given.

The classical Erdős-Ko-Rado (EKR) Theorem states that if we choose a family of subsets, each of size k, from a fixed set of size n (n > 2k), then the largest possible pairwise intersecting family has size t = n−1 k−1 . We consider the probability that a randomly selected family of size t = t n has the EKR property (pairwise nonempty intersection) as n and k = k n tend to infinity, the latter at a specific rate. As t gets large, the EKR property is less likely to occur, while as t gets smaller, the EKR property is satisfied with high probability. We derive the threshold value for t using Janson's inequality. Using the Stein-Chen method we show that the distribution of X 0 , defined as the number of disjoint pairs of subsets in our family, can be approximated by a Poisson distribution. We extend our results to yield similar conclusions for X i , the number of pairs of subsets that overlap in exactly i elements. Finally, we show that the joint distribution (X 0 , X 1 , . . . , X b ) can be approximated by a multidimensional Poisson vector with independent components.

A Sidon set is a set of the positive integers such that the sums of two pairs is not repeated. I. Ruzsa gave a probabilistic construction of an infinite Sidon set. In this work we present the details of a simplified proof of this construction as suggested in a paper of I. Ruzsa and J. Cilleruelo (Real and -padic Sidon sequences, Acta Sci. Math (Szeged) 70 (2004), 505-510).

A Sidon set is a set of the positive integers such that the sums of two pairs is not repeated. I. Ruzsa gave a probabilistic construction of an infinite Sidon set. In this work we present the details of a simplified proof of this construction as suggested in a paper of I. Ruzsa and J. Cilleruelo (Real and -padic Sidon sequences, Acta Sci. Math (Szeged) 70 (2004), 505-510).

A Sidon set is a set of the positive integers such that the sums of two pairs is not repeated. I. Ruzsa gave a probabilistic construction of an infinite Sidon set. In this work we present the details of a simplified proof of this construction as suggested in a paper of I. Ruzsa and J. Cilleruelo (Real and -padic Sidon sequences, Acta Sci. Math (Szeged) 70 (2004), 505-510).

A Sidon set is a set of the positive integers such that the sums of two pairs is not repeated. I. Ruzsa gave a probabilistic construction of an infinite Sidon set. In this work we present the details of a simplified proof of this construction as suggested in a paper of I. Ruzsa and J. Cilleruelo (Real and -padic Sidon sequences, Acta Sci. Math (Szeged) 70 (2004), 505-510).

A Sidon set is a set of the positive integers such that the sums of two pairs is not repeated. I. Ruzsa gave a probabilistic construction of an infinite Sidon set. In this work we present the details of a simplified proof of this construction as suggested in a paper of I. Ruzsa and J. Cilleruelo (Real and -padic Sidon sequences, Acta Sci. Math (Szeged) 70 (2004), 505-510).

A Sidon set is a set of the positive integers such that the sums of two pairs is not repeated. I. Ruzsa gave a probabilistic construction of an infinite Sidon set. In this work we present the details of a simplified proof of this construction as suggested in a paper of I. Ruzsa and J. Cilleruelo (Real and -padic Sidon sequences, Acta Sci. Math (Szeged) 70 (2004), 505-510).