Georges Grekos

On the Erdős–Turán conjecture

Abstract We give equivalent formulations of the Erdős–Turan conjecture on the unboundedness of the number of representations of the natural numbers by additive bases of order two of N . These formulations allow for a quantitative exploration of the conjecture. They are expressed through some functions of x∈ N reflecting the behavior of bases up to x. We examine some properties of these functions and give numerical results showing that the maximum number of representations by any basis is ⩾6.

On densities and gaps

Zusammenfassung Fur jede Menge A ⫅ N sei A ( n ) die Anzahl der Elemente von A ⌢ [1,n] . Es werden die Dichten d A = lim n → ∞ inf A (n) n , d A = lim n → ∞ SUP A (n) n definiert und verschiedene Eigenschaften der “Dichtemenge” S(A) = {( d B, d B) ∈ R 2 |B ⫅ A} bewiesen. Eine besondere Rolle spielt dabei der Anstieg der Randkurve von S ( A ) im Nullpunkt.

VARIATIONS ON A THEME OF CASSELS FOR ADDITIVE BASES

We introduce the notion of caliber, cal(A, B), of a strictly increasing sequence of natural numbers A with respect to another one B, as the limit inferior of the ratio of the nth term of A to that of B. We further consider the limit superior t(A) of the average order of the number of representations of an integer as a sum of two elements of A. We give some basic properties of each notion and we relate the two together, thus yielding a generalization, of the form t(A) ≤ t(B)/cal(A, B), of a result of Cassels specific to the case where A is an additive basis of the natural numbers and B is the sequence of perfect squares. We also provide some formulas for the computation of t(A) in a large class of cases, and give some examples.

Supremum of Representation Functions

Abstract For a subset A of ℕ = {0, 1, 2, . . .}, the representation function of A is defined by r A (n) = |{(a, b) ∈ A × A : a + b = n}|, for n ∈ ℕ, where |E| denotes the cardinality of a set E. Its supremum is the element s(A) = sup{rA (n) : n ∈ ℕ} of . Interested in the question “when is s(A) = ∞?”, we study some properties of the function A ↦ s(A), determine its range, and construct some subsets A of ℕ for which s(A) satisfies certain prescribed conditions.

Open Problems on Densities II

This is a collection of open questions and problems concerning various density concepts on subsets of N: = {1,2,3,…}. It is a continuation of paper [10].

On the General Erdős-Turán Conjecture

The general Erdős-Turán conjecture states that if is an infinite, strictly increasing sequence of natural numbers whose general term satisfies , for some constant and for all , then the number of representations functions of is unbounded. Here, we introduce the function , giving the minimum of the maximal number of representations of a finite sequence of natural numbers satisfying for all . We show that is an increasing function of and that the general Erdős-Turán conjecture is equivalent to . We also compute some values of . We further introduce and study the notion of capacity, which is related to the function by the fact that is the capacity of the set of squares of positive integers, but which is also of intrinsic interest.

Open Problems on Densities

This is a partial and subjective collection of questions and problems concerning various density concepts of subsets of ℕ = {1,2,3,…}.

On the upper and lower exponential density functions

Abstract In the present paper we introduce the upper and lower exponential density functions of subsets A ⊆ ℕ*. We identify completely the form of the upper density and find many properties for the lower one. We provide examples and list some open problems.

Some generalizations of Olivier's theorem

Let ∞ ∑ n=1 an be a convergent series of positive real numbers. L. Olivier proved that if the sequence (an) is non-increasing then lim n→∞ nan = 0. In the present paper: (a) We formulate and prove a necessary and sufficient condition for having lim n→∞ nan = 0; Olivier’s theorem is a consequence of our Theorem 2.1. (b) We prove properties analogous to Olivier’s property when usual convergence is replaced by I-convergence, that is convergence according to an ideal I of subsets of N. Again Olivier’s theorem is a consequence of our Theorem 3.1, when one takes as I the ideal of all finite subsets of N.

Integral Points on a Very Flat Convex Curve

The second named author studied in 1988 the possible relations between the length \(\ell \), the minimal radius of curvature r and the number of integral points N of a strictly convex flat curve in \(\mathbb {R}^2\), stating that \(N = O(\ell /r^{1/3})\) (*), a best possible bound even when imposing the tangent at one extremity of the curve; here flat means that one has \(\ell = r^{\alpha } \) for some \(\alpha \in [2/3, 1)\). He also proved that when \(\alpha \le 1/3\), the quantity N is bounded. In this paper, the authors prove that in general the bound (*) cannot be improved for very flat curves, i.e. those for which \(\alpha \in (1/3, 2/3)\); however, if one imposes a 0 tangent at one extremity of the curve, then (*) is replaced by the sharper inequality \(N \le \ell ^2/r +1\).