On the Erdös-Turán conjecture and related results

The Erdös-Turán Conjecture, posed in 1941 in, states that if a subset B of natural numbers is such that every positive integer n can be written as the sum of a bounded number of terms from B, then the number of such representations must be unbounded as n tends to infinity. The case for h = 2 was given a positive answer by Erdös in 1956. The case for arbitrary h was given by Erdös and Tetali in 1990. Both of these proofs use the probabilistic method, and so the result only shows the existence of such bases but such bases are not given explicitly. Kolountzakis gave an effective algorithm that is polynomial with respect to the digits of n to compute such bases. Borwein, Choi, and Chu showed that the number of representations cannot be bounded by 7. Van Vu showed that the Waring bases contain thin sub-bases. We will discuss these results in the following work

An Effective Additive Basis for the Integers

We give an algorithm for the enumeration of a set E of nonnegative integers with the property that each nonnegative integer x can be written as a sum of two elements of E in at least C 1 log x and at most C 2 log x ways, where C 1 ; C 2 are positive constants. Such a set is called a basis and its existence has been established by Erdos. Our algorithm takes time polynomial in n to enumerate all elements of E not greater than n. We accomplish this by derandomizing a probabilistic proof which is slightly different than that given by Erdos. Mathematics Subject Classification: 11Y16, 68Q99 1 Introduction A set E of nonnegative integers is called a basis if every nonnegative integer can be written as a sum of two elements of E. We write r(x) = r E (x) for the number of representations of x as a + b, with a; b 2 E and a b. In what follows C denotes an arbitrary positive constant, not necessarily the same in all its occurences, and N = f1; 2; 3; : : :g denotes the set of all positive inte..