ON THE ERDŐS-TURÁN ADDITIVE BASE CONJECTURE

In this paper we formulate and prove several variants of the ErdősTurán additive bases conjecture.


Introduction and Problem Statement
Let S be a subset of the natural numbers N and k ∈ N be fixed. Then S is said to be an additive base of order k if every natural number can be expressed as a sum of k elements of S. The weak Goldbach conjecture suggests the set of prime numbers is an additive base of order three [1]. The Erdős-Turán additive bases conjecture is the assertion that all additive bases qualifies very sufficiently to be an additive additive base in the very large. In particular we have the following conjecture of Erdős and Turán (see [2]) This conjecture has garnered the attention of many authors but remains unresolved [3]. By introducing the language of Circles of Partition and associated statistics we reformulate the conjecture in the following manner We exploiting the notion of the density of circles of partition, the notion of ascending, descending and stationary circles of partition and the l th fold energy of circle of partitions to study the Erdős-Turán additive bases conjecture.

The Circle of Partition
In this section we introduce the notion of the circle of partition. We study this notion in-depth and explore some potential applications in the following sequel.
Definition 2.1. Let n ∈ N and M ⊂ N. We denote with the Circle of Partition generated by n with respect to the subset M. We will abbreviate this in the further text as CoP. We call members of C(n, M) as points and denote them by [x]. For the special case M = N we denote the CoP shortly as C(n). then the following relations hold The above language in many ways could be seen as a criterion determining the plausibility of carrying out a partition in a specified set. Indeed this feasibility is trivial if we take the set M to be the set of natural numbers N. The situation becomes harder if we take the set M to be a special subset of natural numbers N, as the corresponding CoP C(n, M) may not always be non-empty for all n ∈ N.
One archetype of problems of this flavour is the binary Goldbach conjecture, when we take the base set M to be the set of all prime numbers P. One could imagine the same sort of difficulty if we extend our base set to other special subsets of the natural numbers. Remark 2.3. It is important to notice that a typical CoP need not have a center. In the case of an absence of a center then we say the circle has a deleted center. It is easy to see that the CoP C(n) contains all points whose weights are positive integers from 1 to n − 1 inclusive: Therefore the CoP C(n) has n−1 2 different axes.
In the sequel we will denote the assignment of an axis L is also an axis with z = y. Then it follows by Definition 2.2 that we must have n = x + y = x + z and therefore y = z. This cannot be and the claim follows immediately. [x] + [y] = n. This contradiction to the Definition 2.1. Due to Proposition 2.4 the case of more than one axis partners is impossible. This completes the proof.

The Density of Points on the Circle of Partition
In this section we introduce the notion of density of points on CoP C(n, M) for M ⊆ N. We launch the following language in that regard. We exploit this notion in a careful manner to study the Erdős-Turán additive bases conjecture.

Ascending, Descending and Stationary Circles of Partition
In this section we introduce the notion of ascending, descending and stationary CoPs between generators. We formalize this notion in the following language. We exploit this notion to improve on the result concerning the Erdős-Turán additive bases conjecture in section 2. We say it is globally ascending (resp. descending) if at ∀m ∈ N it is ascending (resp. descending). We say the CoP C(n, M) is stationary from n to the spot m if for n < m then ν(n, M) = ν(m, M).
Similarly, we say it is globally stationary if it is stationary at all spots m ∈ N. If the CoP C(n, M) is neither globally ascending, descending nor stationary, then we say it is globally oscillatory. Proof. Let C(n, H) be a CoP and assume to the contrary that there are finitely many spots at which it is ascending. Let us name and arrange the spots as follows m 1 < m 2 < · · · < m k . It follows that for all i ≥ 1. The upshot is that This, however, violates the requirement of the statement, thereby ending the proof.

The l th Fold Energy of Circles of Partition
In this section we introduce and study the notion of the l th fold energy of CoPs and exploit some applications in this context. This notion tends to more effective and extends very much to sequences not necessarily having a positive density. for a fixed l ∈ N.
It is important to remark that the l th energy of a typical CoP C(n, M) could either be infinite or finite. In that latter case it certainly should have a finite value. To that effect we state the following proposition.
Proposition 5.2. Let J l ⊂ N be the set of all l th powers. Then E(l, J l ) < ∞ for all l ≥ 3 and E(2, J 2 ) = ∞.
Proof. Let l ≥ 3 be fixed and consider the CoP C(n l , J l ), where J l ⊂ N is the set of all l th powers. Then it follows from the configuration of CoPs the following inequality  Proof. First we compute the two fold energy E(2, B) of the CoP C(n, B). Since G B (n) > 0 for all sufficiently large values of n, it follows that G B (n 2 ) > 0 for all sufficiently large values of n so that for all k large enough there exist some constant L = L(k) > 0 such that we can write Let B be an additive base of order 2, then it is well-known that # {n ≤ x| n ∈ B} ≥ √ x.
In line with this tied with Theorem 5.4 the solution to the Erdős-Turán additive bases conjecture is an easy consequence. 1 .