Erdős–Tetali theorem explained

In additive number theory, an area of mathematics, the Erdős–Tetali theorem is an existence theorem concerning economical additive bases of every order. More specifically, it states that for every fixed integer

h\geq2

, there exists a subset of the natural numbers

l{B}\subseteqN

satisfyingr_(n) \asymp \log n,where

rl{B,h}(n)

denotes the number of ways that a natural number n can be expressed as the sum of h elements of B.[1]

The theorem is named after Paul Erdős and Prasad V. Tetali, who published it in 1990.

Motivation

The original motivation for this result is attributed to a problem posed by S. Sidon in 1932 on economical bases. An additive basis

l{B}\subseteqN

is called economical[2] (or sometimes thin[3]) when it is an additive basis of order h and

rl{B,h}(n)\ll\varepsilonn\varepsilon

for every

\varepsilon>0

. In other words, these are additive bases that use as few numbers as possible to represent a given n, and yet represent every natural number. Related concepts include

Bh[g]

-sequences
[4] and the Erdős–Turán conjecture on additive bases.

Sidon's question was whether an economical basis of order 2 exists. A positive answer was given by P. Erdős in 1956,[5] settling the case of the theorem. Although the general version was believed to be true, no complete proof appeared in the literature before the paper by Erdős and Tetali.[6]

Ideas in the proof

The proof is an instance of the probabilistic method, and can be divided into three main steps. First, one starts by defining a random sequence

\omega\subseteqN

by

\Pr(n\in\omega)=C

1-1
h
n

(log

1
h
n)

,

where

C>0

is some large real constant,

h\geq2

is a fixed integer and n is sufficiently large so that the above formula is well-defined. A detailed discussion on the probability space associated with this type of construction may be found on Halberstam & Roth (1983).[7] Secondly, one then shows that the expected value of the random variable

r\omega,h(n)

has the order of log. That is,

E(r\omega,h(n))\asymplogn.

Finally, one shows that

r\omega,h(n)

almost surely concentrates around its mean. More explicitly:

\Pr(\existsc1,c2>0~|~foralllargen\inN,~c1E(r\omega,h(n))\leqr\omega,h(n)\leqc2E(r\omega,h(n)))=1

This is the critical step of the proof. Originally it was dealt with by means of Janson's inequality, a type of concentration inequality for multivariate polynomials. Tao & Vu (2006)[8] present this proof with a more sophisticated two-sided concentration inequality by V. Vu (2000),[9] thus relatively simplifying this step. Alon & Spencer (2016) classify this proof as an instance of the Poisson paradigm.[10]

Relation to the Erdős–Turán conjecture on additive bases

See main article: Erdős–Turán conjecture on additive bases. The original Erdős–Turán conjecture on additive bases states, in its most general form, that if \mathcal\subseteq\mathbb is an additive basis of order h then

\limsupn\torl{B,h}(n)=infty;

that is, r_(n) cannot be bounded. In his 1956 paper, P. Erdős[5] asked whether it could be the case that

\limsupn\toinfty

rl{B,2
(n)}{log

n}>0

whenever

l{B}\subseteqN

is an additive basis of order 2. In other words, this is saying that r_(n) is not only unbounded, but that no function smaller than log can dominate r_(n). The question naturally extends to

h\geq3

, making it a stronger form of the Erdős–Turán conjecture on additive bases. In a sense, what is being conjectured is that there are no additive bases substantially more economical than those guaranteed to exist by the Erdős–Tetali theorem.

Further developments

Computable economical bases

l{R}\subseteqN

satisfying

rl{R,2}(n)\asymplogn

such that

l{R}\cap\{0,1,\ldots,n\}

takes polynomial time in n to be computed. The question for

h\geq3

remains open.

Economical subbases

Given an arbitrary additive basis

l{A}\subseteqN

, one can ask whether there exists

l{B}\subseteql{A}

such that

l{B}

is an economical basis. V. Vu (2000)[12] showed that this is the case for Waring bases

N\wedgek:=\{0k,1k,2k,\ldots\}

, where for every fixed k there are economical subbases of

N\wedgek

of order

s

for every

s\geqsk

, for some large computable constant

sk

.

Growth rates other than log

Another possible question is whether similar results apply for functions other than log. That is, fixing an integer

h\geq2

, for which functions f can we find a subset of the natural numbers

l{B}\subseteqN

satisfying

rl{B,h}(n)\asympf(n)

? It follows from a result of C. Táfula (2019)[13] that if f is a locally integrable, positive real function satisfying
1
x
x
\int
1

f(t)dt\asympf(x)

, and

logx\llf(x)\ll

1
h-1
x

(logx)-\varepsilon

for some

\varepsilon>0

,then there exists an additive basis

l{B}\subseteqN

of order h which satisfies

rl{B,h}(n)\asympf(n)

. The minimal case recovers Erdős–Tetali's theorem.

See also

C\inR

, there is no set

l{B}\subseteqN

which satisfies

style{\sumn\leqrl{B,2}(n)=Cx+o\left(x1/4log(x)-1/2\right)}

.

l{B}\subseteqN

is an additive basis of order 2, then

rl{B,2}(n)O(1)

.

k\geq2

.

References

Notes and References

  1. Alternative statement in big Theta notation: r_(n) = \Theta(\log(n)),
  2. As in Halberstam & Roth (1983), p. 111.
  3. As in Tao & Vu (2006), p. 13.
  4. See Definition 3 (p. 3) of O'Bryant, K. (2004), "A complete annotated bibliography of work related to Sidon sequences", Electronic Journal of Combinatorics, 11: 39.
  5. Erdős. P.. 1956. Problems and results in additive number theory. Colloque sur la Théorie des Nombres. 127–137.
  6. See p. 264 of Erdős–Tetali (1990).
  7. See Theorem 1 of Chapter III.
  8. Section 1.8 of Tao & Vu (2006).
  9. Vu. Van H.. 2000-07-01. On the concentration of multivariate polynomials with small expectation. Random Structures & Algorithms. en. 16. 4. 344–363. 10.1002/1098-2418(200007)16:4<344::aid-rsa4>3.0.co;2-5. 1098-2418. 10.1.1.116.1310.
  10. Chapter 8, p. 139 of Alon & Spencer (2016).
  11. Kolountzakis. Mihail N.. 1995-10-13. An effective additive basis for the integers. Discrete Mathematics. 145. 1. 307–313. 10.1016/0012-365X(94)00044-J. free.
  12. Vu. Van H.. 2000-10-15. On a refinement of Waring's problem. Duke Mathematical Journal. 105. 1. 107–134. 10.1215/s0012-7094-00-10516-9. 0012-7094. 10.1.1.140.3008.
  13. Táfula. Christian. An extension of the Erdős-Tetali theorem. Random Structures & Algorithms. en. 55. 1. 173–214. 10.1002/rsa.20812. 1098-2418. 2019. 1807.10200. 119249787 .