Charles Helou

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.

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.

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.

On Wendt's determinant

Wendts determinant of order m is the circulant determinant Wm whose (i,j)-th entry is the binomial coefficient ( m |i-j| ), for 1 ≤ i,j ≤ m. We give a formula for W m when m is even not divisible by 6, in terms of the discriminant of a polynomial T m+1 , with rational coefficients, associated to (X + 1) m+1 - X m+1 - 1. In particular, when m = p - 1 where p is a prime = -1 (mod 6), this yields a factorization of W p-1 involving a Fermat quotient, a power of p and the 6-th power of an integer.

ANALYTIC ERDÖS-TURÁN CONJECTURES AND ERDÖS-FUCHS THEOREM

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 Erdos-Turan conjectures in terms of supported series and extending to them a version of Erdos-Fuchs theorem.

Additive Bases Representations and the Erdős-Turán Conjecture

We give a lower bound to the maximal number of representations by an additive basis of the natural numbers, in conjunction with a celebrated conjecture of Erdős and Turan.