On The Erdös-Turán Additive Base Conjecture
2021
https://doi.org/10.21203/RS.3.RS-660255/V1…
8 pages
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Abstract
In this paper we formulate and prove several variants of the Erdös-Turán additive bases conjecture.
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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.

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References (3)
- Helfgott, Harald A, The ternary Goldbach conjecture is true, arXiv preprint arXiv:1312.7748, 2013.
- Erdos, Paul and Turán, Pál, Unsolved problems in number theory, J. London Math. Soc, vol.16(4),1941 , 212-215.
- Tao, Terence and Vu, Van H, Additive combinatorics, Cambridge University Press, vol.105, 2006.