Erdős–Turán conjecture on additive bases
"(2) If for , then "
Here is the number of ways one can write the natural number as the sum of two (not necessarily distinct) elements of . If is always positive for sufficiently large , then is called an additive basis (of order 2). This problem has attracted significant attention but remains unsolved.
In 1964, Erdős published a multiplicative version this conjecture. See source :
- P. Erdõs: On the multiplicative representation of integers, Israel J. Math. 2 (1964), 251--261
While the conjecture remains unsolved, there have been some advances on the problem. First, we express the problem in modern language. For a given subset , we define its representation function . Then the conjecture states that if for all sufficiently large, then .
More generally, for any and subset , we can define the representation function as . We say that is an additive basis of order if for all sufficiently large. One can see from an elementary argument that if is an additive basis of order , then
So we obtain the lower bound .
The original conjecture spawned as Erdős and Turán sought a partial answer to Sidon's problem (see: Sidon sequence). Later, Erdős set out to answer the following question posed by Sidon: how close to the lower bound can an additive basis of order get? This question was answered positively in the case by Erdős in 1956. Erdős proved that there exists an additive basis of order 2 and constants such that for all sufficiently large. In particular, this implies that there exists an additive basis such that , which is essentially best possible. This motivated Erdős to make the following conjecture
If is an additive basis of order , then
In 1986, Eduard Wirsing proved that a large class of additive bases, including the prime numbers, contains a subset that is an additive basis but significantly thinner than the original. In 1990, Erdős and Tetali extended Erdős's 1956 result to bases of arbitrary order. In 2000, V. Vu proved that thin subbases exist in the Waring bases using the Hardy–Littlewood circle method and his polynomial concentration results. In 2006, Borwein, Choi, and Chu proved that for all additive bases , eventually exceeds 7. 
- Erdős, Paul.; Turán, Pál (1941). "On a problem of Sidon in additive number theory, and on some related problems". Journal of the London Mathematical Society 16: 212–216. doi:10.1112/jlms/s1-16.4.212.
- Tao, T.; Vu, V. (2006). Additive Combinatorics. New York: Cambridge University Press. p. 13. ISBN 0-521-85386-9.
- Erdős, P. (1956). "Problems and results in additive number theory". Colloque sur le Theorie des Nombres: 127–137.
- Wirsing, Eduard (1986). "Thin subbases". Analysis 6: 285–308. doi:10.1524/anly.19188.8.131.525.
- Erdős, Paul.; Tetalli, Prasad (1990). "Representations of integers as the sum of terms". Random Structures Algorithms 1 (3): 245–261. doi:10.1002/rsa.3240010302.
- Vu, Van (2000). "On a refinement of Waring's problem". Duke Mathematical Journal 105 (1): 107–134. doi:10.1215/S0012-7094-00-10516-9.
- Borwein, Peter; Choi, Stephen; Chu, Frank (2006). "An old conjecture of Erdős–Turán on additive bases". Mathematics of Computation 75: 475–484. doi:10.1090/s0025-5718-05-01777-1.
- Xiao, Stanley Yao (2011). On the Erdős–Turán conjecture and related results.