Erdős–Turán conjecture on additive bases
The Erdős–Turán conjecture is an old unsolved problem in additive number theory (not to be confused with Erdős conjecture on arithmetic progressions) posed by Paul Erdős and Pál Turán in 1941.
History
The conjecture was made jointly by Paul Erdős and Pál Turán in.[1] In the original paper, they state
"(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).[2] This problem has attracted significant attention[2] 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
Progress
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.[3] 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.[4] In 1990, Erdős and Prasad V. Tetali extended Erdős's 1956 result to bases of arbitrary order.[5] In 2000, V. Vu proved that thin subbases exist in the Waring bases using the Hardy–Littlewood circle method and his polynomial concentration results.[6] In 2006, Borwein, Choi, and Chu proved that for all additive bases ,
eventually exceeds 7.[7]
[8]
References
- ↑ 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.
- ↑ 2.0 2.1 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.1986.6.23.285.
- ↑ Erdős, Paul.; Tetali, 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.