Some remarks on the Erd˝ os–Tur´ an conjecture

LXIII.4 (1993)

Some remarks on the Erd˝ os–Tur´ an conjecture

by

Martin Helm (Mainz)

Notation. In additive number theory an increasing sequence of natural numbers is called an asymptotic basis of order h of N if every sufficiently large n ∈ N can be written as the sum of h elements of A.

Let r n (h, A) denote the number of representations of n as n = a 1 +. . .+a h

with a 1 , . . . , a h ∈ A and a ₁ ≤ . . . ≤ a _h .

If A satisfies r n (h, A) ≤ g ∀n ∈ N, where g is a natural constant, then A is called a B h [g]-sequence (and in the special case g = 1 a B h -sequence).

Furthermore, for any given sequence A of natural numbers and any m ∈ N we define

δ A (m) := |{(a i , a j ) : a i , a j ∈ A, m = a _j − a _i }|, and for a given N ∈ N,

h A (m) := |{(a i , a j ) : a i , a j ∈ A ∩ [1, N ² ], m = a j − a _i }|.

Introduction. A famous conjecture of Erd˝ os and Tur´ an [2] asserts that there exists no asymptotic basis of order 2 of N that is a B 2 [g]-sequence at the same time. Erd˝ os shows (see [4]) that if A is an arbitrary sequence of natural numbers satisfying lim inf n→∞ A(n)/ √

n > 0 and N is a given natural number then

(1) H A (N ) :=

N

X

m=1

h A (m)  N log N , which proves the above hypothesis in the special case g = 1.

Almost all known results on B h [g]-sequences are based on considerations concerning the representation of certain natural numbers as a difference of elements of a given sequence A.

Therefore for a further proof of the Erd˝ os–Tur´ an conjecture it is very

interesting to decide whether Erd˝ os’ estimate (1) is sharp with respect to

magnitude. Here we prove by means of an explicit construction that (1) is

indeed sharp in the above sense.

Furthermore, (1) unfortunately does not render possible an estimation of h A (m) for any specific m ∈ [1, N ] but only provides some average in- formation; in particular, it is not possible to decide whether any specific m ∈ [1, N ] for a given N satisfies e.g. h A (m) > c log N or not, where c is a constant. Here we prove — again by means of an explicit construction

— the existence of two increasing sequences of natural numbers B and M satisfying

lim inf

n→∞

B(n) √

n > 0, lim inf

n→∞

M (n) log n > 0 and

δ B (m j ) ≡ 1 ∀j ≥ j ₀ .

Theorem. There exists an infinite sequence of natural numbers A sat- isfying

(2) lim inf

n→∞

A(n) √ n > 0 and

(3) H A (N ) =

N

X

m=1

h A (m)  N log N .

P r o o f. We will prove the above theorem by constructing an infinite sequence A of natural numbers as an infinite countable union of finite B 2 - sets (also called Sidon sets).

Let µ and ν be arbitrary natural numbers satisfying ν > 2 and µ > 1.

Now a sequence (n j ) _j∈N is defined inductively as follows:

n 1 := 1,

n 2j := µn 2j−1 , j ∈ N , n 2j+1 := νn 2j , j ∈ N . Therefore

n 2j = µ ^j ν ^j−1 , j ∈ N , n 2j+1 = µ ^j ν ^j , j ∈ N ⁰ . We define

I k := ]n k−1 , n k [ ∀k ≥ 2 .

A well-known result of Erd˝ os and Chowla [1], [2] states that

(4) lim inf

n→∞

F 2 (n)

√ n ≥ 1 ,

where F 2 (n) denotes the maximum number of elements that can be selected from the set 1, 2, . . . , n to form a B 2 -sequence. Since the “B 2 -property” of a finite set is invariant under translations, for any j ∈ N (4) proves the existence of a Sidon set S 2j ⊆ I _2j = ]n 2j−1 , n 2j [ such that

(5) |S _2j |  pn 2j − n _2j−1 = p

µ − 1 √

n 2j−1 (j → ∞).

We define

A :=

∞

[

j=1

S 2j

and have to show that A satisfies the conditions (2) and (3).

P r o o f o f (2). For any m ∈ N with m > µ there exists a j 0 ∈ N with n 2j

≤ m < n _2j

₊₂ .

Therefore

√ m < √

n 2j

+2  √

n 2j

−1 , and on the other hand,

A(m) ≥ A(n 2j

) − A(n 2j

−1 ) = |S 2j

|  √

n 2j

−1 . Thus lim inf n→∞ A(n)/ √

n > 0 and (2) holds.

P r o o f o f (3). For a given N ∈ N with N > µ there exists a j 1 ∈ N such that

n 2j

−2 ≤ N < n 2j

and for any m ∈ [1, N ] we define

h ¹ _A (m) := |{(a i , a k ) : a i , a k ∈ [1, n _2j

] ∩ A and m = a k − a _i }|, h ² _A (m) := |{(a i , a k ) : a i ∈ [1, N ² ] ∩ A, a k ∈ ]n _2j

, N ² ] and m = a k − a _i }|,

H _A ¹ (N ) :=

N

X

m=1

h ¹ _A (m), H _A ² (N ) :=

N

X

m=1

h ² _A (m).

Consequently,

h A (m) = h ¹ _A (m) + h ² _A (m), H A (N ) = H _A ¹ (N ) + H _A ² (N ).

E s t i m a t i o n o f H _A ¹ (N ). Obviously

H _A ¹ (N ) < (A(n 2j

)) ²  n _2j

 N .

E s t i m a t i o n o f H _A ² (N ). Since ν > 2, for any j ≥ j 1 the length of the gap between two consecutive Sidon sets S 2j+2 and S 2j is bigger than n 2j

+1 − n _2j

> n 2j

> N .

Therefore a number m ∈ [1, N ] can be represented as a difference of two

elements a i , a k of A with a k > n 2j

only if a k and a i are elements of the

same Sidon subset of A.

Let Θ N

be the number of Sidon subsets S 2j of A satisfying S 2j ∩ [1, N ² ] 6= ∅ .

Then the B 2 -property of all Sidon subsets of A leads to h ² _A (m) ≤ Θ N

∀m ∈ [1, N ] and consequently,

(6) H _A ² (N ) =

N

X

m=1

h ² _A (m) ≤ N Θ N

.

E s t i m a t i o n o f Θ N

. For given N ∈ N with N > µ, there exists a j 2 ∈ N so that

n 2j

−2 ≤ N ² < n 2j

⇒ µ ^j

⁻¹ ν ^j

⁻² ≤ N ² ⇒ j ₂  log N and as Θ N

≤ j ₂

(7) ⇒ Θ _N

 log N .

Thus (6) and (7) imply H _A ² (N )  N log N and

H A (N ) = H _A ¹ (N ) + H _A ² (N )  N log N , which completes the proof.

Corollary. There exist two infinite increasing sequences B and M of natural numbers satisfying

lim inf

n→∞

B(n) √ n > 0 , (8)

lim inf

n→∞

M (n) log n > 0 (9)

and

(10) δ B (m) ≡ 1 ∀m ∈ M .

P r o o f. Let A be the infinite sequence of natural numbers generated by the construction of the above theorem in the special case µ = 7/4 and ν = 4 (where the inductive definition of the sequence (n j ) _j∈N is supplemented by the definition n 2 := 2 which does not restrict at all the applicability of the proof).

Consequently, in this special case we define

n 1 := 1, n 2j+1 := νn 2j , j ∈ N, n 2 := 2, n 2j := µn 2j−1 , j ≥ 2 . Thus

n 2j = 2 · 7 ^j−1 , n 2j+1 = 8 · 7 ^j−1 , j ∈ N .

For any j ∈ N we define

D 2j := {m ∈ N : ∃a i , a j ∈ S _2j and m = a j − a _i } . Since S 2j is a Sidon set and

j→∞ lim

|S _2j |

√ n 2j − n _2j−1 = 1 there exists a j 0 ∈ N so that

|D _2j | =  |S _2j | 2



> n 2j−2 ∀j ≥ j ₀ . Since, on the other hand,

m ∈ D 2j ⇒ m < n 2j − n _2j−1 = 3n 2j−2 ∀j ∈ N , for any j ≥ j 0 there exists at least one m j ∈ D 2j satisfying

n 2j−2 < m j < 3n 2j−2 .

Let M be defined as the sequence m j

, m j

+1 , m j

+2 , . . . Now we will con- struct a subsequence B of A by eliminating a negligible number of elements of A so that B will still satisfy

lim inf

n→∞

B(n) √ n > 0 and δ B (m j ) = 1 will hold for all j ≥ j 0 .

Since, according to the definition,

n 2j−2 < m j ∀j ≥ j ₀ , we have

m j = a k − a _i ⇒ a _k > n 2j−2 . Therefore, since A ∩ ]n 2j−2 , n 2j−1 [ = ∅, a k satisfies (11) a k > n 2j−1 and a k 6∈

j−1

[

h=1

S 2h .

On the other hand, for h ≥ j the length of the gap between two consecutive Sidon subsets S 2h−2 , S 2h is bigger than n 2j−1 − n _2j−2 = 3n 2j−2 . Therefore as according to the definition m j < 3n 2j−2 for any j ≥ j 0 , m j can occur as a difference a k − a _i , a i , a k ∈ A, only if both a _k and a i are elements of the same Sidon subset S 2h , h ≥ j.

Therefore the B 2 -property of the sets S 2j implies that

|D 2j ∩ M | ≤ j ∀j ∈ N .

Thus ∀j ≥ j 0 it is possible to construct a new Sidon set S _2j ⁰ from S 2j by eliminating less than j elements of S 2j so that

D _2j ⁰ ∩ M = {m _j } ,

where

D _2j ⁰ := {m ∈ N : ∃(a i , a k ) ∈ S _2j ⁰ , m = a k − a _i } . We define

B :=

∞

[

j=1

S _2j ⁰ , where S _2j ⁰ := S 2j , 1 ≤ j < j 0 . Obviously δ B (m j ) = 1 ∀j ≥ j 0 and

|S _2j ⁰ |  |S _2j | ⇒ lim inf

n→∞

B(n) √ n > 0 .

Furthermore, Θ n  log n (n → ∞), where Θ n is the number of Sidon subsets S _2j ⁰ of B satisfying S _2j ⁰ ∩ [1, n] 6= ∅. Consequently

lim inf

n→∞

M (n) log n > 0 , which completes the proof.

References

[1] S. C h o w l a, Solution of a problem of Erd˝ os and Tur´ an in additive number theory , Proc. Nat. Acad. Sci. India 14 (1944), 1–2.

[2] P. E r d ˝ o s and P. T u r ´ a n, On a problem of Sidon in additive number theory and some related problems, J. London Math. Soc. 16 (1941), 212–215; Addendum (by P. Erd˝ os), ibid. 19 (1944), 208.

[3] H. H a l b e r s t a m and K. F. R o t h, Sequences, Springer, New York 1983.

[4] A. S t ¨ o h r, Gel¨ oste und ungel¨ oste Fragen ¨ uber Basen der nat¨ urlichen Zahlenreihe. II , J. Reine Angew. Math. 194 (1955), 111–140.

FACHBEREICH MATHEMATIK

JOHANNES GUTENBERG-UNIVERSIT ¨ AT MAINZ SAARSTR. 21

D-6500 MAINZ, GERMANY

Received on 4.8.1992

and in revised form on 24.9.1992 (2289)