The Erdős–Turán property for a class of bases

International Journal of Mathematics and Mathematical Sciences, 2005

We consider and study formal power series, that we call supported series, with real coefficients which are either zero or bounded below by some positive constant. The sequences of such coefficients have a lot of similarity with sequences of natural numbers considered in additive number theory. It is this analogy that we pursue, thus establishing many properties and giving equivalent statements to the well-known Erdös-Turán conjectures in terms of supported series and extending to them a version of Erdös-Fuchs theorem.

Acta Arithmetica, 2005

arXiv: General Mathematics, 2020

In this paper we formulate and prove several variants of the Erdős-Turan additive bases conjecture.

arXiv (Cornell University), 2017

In this paper we use the former of the authors developed theory of circles of partition to investigate possibilities to prove the binary Goldbach as well as the Lemoine conjecture. We state the squeeze principle and its consequences if the set of all odd prime numbers is the base set. With this tool we can prove asymptotic versions of the binary Goldbach as well as the Lemoine conjecture.

Annals of Mathematics, 2003

The basic theme of this paper is the fact that if A is a finite set of integers, then the sum and product sets cannot both be small. A precise formulation of this fact is Conjecture 1 below due to Erdős-Szemerédi [E-S]. (see also [El], [T], and [K-T] for related aspects.) Only much weaker results or very special cases of this conjecture are presently known. One approach consists of assuming the sum set A + A small and then deriving that the product set AA is large (using Freiman's structure theorem) (cf. [N-T], [Na3]). We follow the reverse route and prove that if |AA| < c|A|, then |A + A| > c |A| 2 (see Theorem 1). A quantitative version of this phenomenon combined with the Plünnecke type of inequality (due to Ruzsa) permit us to settle completely a related conjecture in [E-S] on the growth in k. If g(k) ≡ min{|A[1]| + |A{1}|} over all sets A ⊂ Z of cardinality |A| = k and where A[1] (respectively, A{1}) refers to the simple sum (resp., product) of elements of A. (See (0.6), (0.7).) It was conjectured in [E-S] that g(k) grows faster than any power of k for k → ∞. We will prove here that ln g(k) ∼ (ln k) 2 ln ln k (see Theorem 2) which is the main result of this paper.

2004

The basic theme of this paper is the fact that if A is a finite set of integers, then the sum and product sets cannot both be small. A precise formulation of this fact is Conjecture 1 below due to Erdős-Szemerédi [E-S]. (see also [El], [T], and [K-T] for related aspects.) Only much weaker results or very special cases of this conjecture are presently known. One approach consists of assuming the sum set A + A small and then deriving that the product set AA is large (using Freiman’s structure theorem) (cf. [N-T], [Na3]). We follow the reverse route and prove that if |AA | &lt; c|A|, then |A + A |&gt; c ′ |A | 2 (see Theorem 1). A quantitative version of this phenomenon combined with the Plünnecke type of inequality (due to Ruzsa) permit us to settle completely a related conjecture in [E-S] on the growth in k. If g(k) ≡ min{|A[1] | + |A{1}|} over all sets A ⊂ Z of cardinality |A | = k and where A[1] (respectively, A{1}) refers to the simple sum (resp., product) of elements of A. (See (...

2021

In this paper we continue the development of the method of Circles of Partition by introducing the notion of generalized circles of partition. This is an extension program of the notion of circle of partition developed in our first paper [1]. As an application we prove an analogue of the Erdős-Turán additive base conjecture.

Combinatorica, 1995

In this paper new proofs of the Canonical Ramsey Theorem, which originally has been proved by ErdSs and Rado, are given. These yield improvements over the known bounds for the arising Erd6s-Rado numbers ER(k; l), where the numbers ER(k; l) are defined as the least positive integer n such that for every partition of the k-element subsets of a totally ordered n-element set X into an arbitrary number of classes there exists an /-element subset Y of X, such that the set of kelement subsets of Y is partitioned canonically (in the sense of Erd6s and Rado). In particular, it is shown that 2 cl'12 < ER(2; l) < 2 c2"/2'l~ for every positive integer 1 >_ 3, where Cl,C 2 are positive constants. Moreover, new bounds, lower and upper, for the numbers ER(k;l) for arbitrary positive integers k, l are given. HANNO LEFMANN, VOJTI~CH RODL Theorem 1.2. [14], [10], [9] Let k, t be positive integers with k > 3 and t >_ 2. * such that there exist positive constants ck,t, ck, t and Then

Journal of Combinatorial Theory, Series A, 2004

We show that the analogue of the Erdo +s-Tura´n conjecture, for the number of representations by a basis of order two of an additive semi-group, does not hold in a variety of additive groups derived from those of certain fields. This is done by explicitly constructing some bases for which we estimate the maximal number of representations of the elements of the group as a sum of two elements from the given basis.

Journal of Combinatorial Theory, 1966

The number of members of a family of sequences of length n of zeros and ones whose members differ from one another in at most 2 k places is shown to be necessarily no more than y~. A generalization to a similar statement about families of sequences of integers, and the relation of this theorem to a result of Katona are also described.