C. Baxa and J. Schoissengeier (Wien)

LXVIII.3 (1994)

Minimum and maximum order of magnitude of the discrepancy of (nα)

by

C. Baxa and J. Schoissengeier (Wien)

Dedicated to Prof. Wolfgang Schmidt on the occasion of his sixtieth birthday

It is a classical result of P. Bohl [5], W. Sierpiński [21, 22] and H. Weyl [25, 26] that the sequence (nα) n≥1 is uniformly distributed modulo 1 if and only if α is irrational. The discrepancies

D _N ^∗ (α) = sup

0≤x≤1

X N n=1

c _[0,x) ({nα}) − N x    and

D _N (α) = sup

0≤x<y≤1

X N n=1

c _[x,y) ({nα}) − N (y − x)   

measure the deviation of this sequence from an ideal distribution. (Here N ∈ N, c M is the characteristic function of the set M and {x} = x − [x]

denotes the fractional part of x.) The speed of convergence in the limit relations

N →∞ lim 1

N D ^∗ _N (α) = 0 and lim

N →∞

1

N D _N (α) = 0

is used as a measure for the quality of distribution and was studied by many authors. Initially the problem was tackled by H. Behnke [3, 4], A. Ostrowski [14], G. H. Hardy and J. E. Littlewood [10], and E. Hecke [11]. More re- cently, it was taken up by H. Niederreiter [13], J. Lesca [12], V. T. S´os [23, 24], Y. Dupain [7, 8], Y. Dupain and V. T. S´os [9], L. Ramshaw [15] and J. Schoißengeier [17, 18, 20].

Research was supported by the Austrian Science Foundation (FWF) under grant P8703–PHY.

[281]

In [17] it was proved that lim inf N →∞ D ^∗ _N (α) = 1 for all irrational α. To determine the maximum order of D ^∗ _N (α), the quantities

ω _N ⁺ (α) = sup

0≤x≤1

 X ^N

n=1

c _[0,x) ({nα}) − N x



and

ω _N ⁻ (α) = sup

0≤x≤1

 N x −

X N n=1

c _[0,x) ({nα})



were introduced by J. Schoißengeier [20] who determined

1≤N <q max

ω ⁺ _N (α) and max

1≤N <q

ω _N ⁻ (α)

up to an absolute error in terms of the continued fraction expansion of α.

Utilizing D ^∗ _N (α) = max(ω ⁺ _N (α), ω ⁻ _N (α)) one arrives at the maximum order of D _N ^∗ (α).

It is the purpose of this paper to prove analogous results for the minimum and maximum order of D N (α). We calculate max 1≤N <q

D N (α) in terms of the continued fraction expansion of α up to an absolute error (where q _m denotes the denominator of the mth convergent of α). Using this we describe the maximum order of the sequence (D N (α)) N ≥1 and calculate lim sup _{N →∞} D _N (α)/ log N for all α for which D _N (α) = O(log N ) is satisfied.

Finally, we determine the minimum order of (D _N (α)) _{N ≥1} which turns out to be closely connected to the Lagrange spectrum.

1. The maximum order. We will use the following notations: α will always denote an irrational real number with regular continued fraction expansion α = [a 0 , a 1 , a 2 , . . .] (a 0 ∈ Z and a 1 , a 2 , . . . ∈ N) and convergents (p _m /q _m ) _m≥0 . For all i, j ≥ 0 let

s ij = q _min(i,j) (q _max(i,j) α − p _max(i,j) ) and

ε _i = 1

2 (1 − (−1) ^a

) Y

0≤j≤i j≡i (mod 2)

(−1) ^a

.

We are now prepared to state our first main result.

Theorem 1.1. For m ≥ 0 let N m = ¹ ₂ P _m

i=0 (a i+1 + (−1) ^m ε i )q i . Then as m → ∞,

4 max

1≤N <q

D _N (α) = X m i=0

a _i+1 − X

0≤i≤m

X

0≤j≤m j≡i (mod 2)

ε _i ε _j |s _ij | + O(1)

and

1≤N <q max

D _N (α) =

 D _N

(α) + O(1) if N _m < q _m+1 , D N

(α) + O(1) otherwise.

The implicit constants are absolute.

P r o o f. We introduce S _m = 1

4 X m i=0

a _i+1 − 1 4

X

0≤i≤m

X

0≤j≤m j≡i (mod 2)

ε _i ε _j |s _ij |

as a convenient shorthand notation.

Employing c _[0,{x−y}) ({x}) − {x − y} = {y} − {x} for all x, y ∈ R we have for 0 ≤ k, l ≤ N < q m+1 ,

(∗) ∆ _N (k, l) :=

X N n=1

c _[0,{kα}) ({nα}) − N {kα} − X N n=1

c _[0,{lα}) ({nα}) + N {lα}

= X N n=1

({(n − k)α} − {nα}) − X N n=1

({(n − l)α} − {nα})

=

k−1 X

n=1

{−nα} +

N −k X

n=1

{nα} − X l−1 n=1

{−nα} −

N −l X

n=1

{nα}

= k − 1 −

k−1 X

n=1

{nα} − (l − 1) + X l−1 n=1

{nα} +

N −k X

n=1

{nα} −

N −l X

n=1

{nα}

= X l−1 n=1

({nα}−1/2)+

N −k X

n=1

({nα}−1/2)−

k−1 X

n=1

({nα}−1/2)−

N −l X

n=1

({nα}−1/2)

≤ 2 max

1≤M <q

X M n=1

B ₁ (nα) − 2 min

1≤M <q

X M n=1

B ₁ (nα) = S _m + O(1).

Here B ₁ (x) = {x}−1/2 denotes the first Bernoulli polynomial. The last step made use of Corollary 2 in §2 of [19]. Using D _N (α) = 1+max _1≤k,l≤N ∆ _N (k, l) we get max 1≤N <q

D N (α) ≤ S m + c with an absolute constant c > 0. To obtain equality we set

k := 1 + 1 2

X

0≤i≤m i≡0 (mod 2)

(a _i+1 + (−1) ^m ε _i )q _i ,

l := 1 + 1 2

X

0≤i≤m i≡1 (mod 2)

(a _i+1 + (−1) ^m ε _i )q _i

and b N m := k +l −1 = N m +1. Obviously l −1 = b N m −k and k −1 = b N m −l.

According to Corollary 2 in §2 of [19] we have equality in (∗). Had we proved N b _m < q _m+1 we would have completed the proof of the theorem. It is of no importance that b N _m = N _m + 1 as D _{N +1} (α) = D _N (α) + O(1) with an absolute implied constant. A trivial estimation yields

N _m ≤ 1 2

X m i=0

2a _i+1 q _i = q _m+1 + q _m − 1.

If a _m+1 ≥ 3 we even have

N m < (q m + q m−1 ) + 1

2 (a m+1 + 1)q m ≤ q m+1 .

Thus, N _m ≥ q _m+1 only if a _m+1 ≤ 2. But in this case we may safely change to N _m−1 < q _m + q _m−1 ≤ q _m+1 as S _m = S _m−1 + O(a _m+1 ) with an absolute implied constant.

We conclude the proof with a remark: Obviously N m ≥ q m if a m+1 ≥ 2.

If a _m+1 = a _m = 1 it is possible that a _m+1 + (−1) ^m ε _m = a _m + (−1) ^m−1 ε _m−1

= 0 but by the definition of the ε _i it is impossible to have also a _m−1 + (−1) ^m−2 ε m−2 = 0. Therefore N m ≥ q m−2 .

R e m a r k. Using Corollary 1 in §2 of [19] the Bernoulli polynomials can be replaced by Dedekind sums in the above estimate. This indicates a close connection between discrepancies and Dedekind sums which was first pointed out and explored by U. Dieter (oral communication).

Corollary 1.2. Let α be an irrational number. For N ∈ N we define m ∈ N by the property q m ≤ N < q m+1 . Then

lim sup

N →∞

D N (α)

.  X ^m

i=0

a i+1 − X

0≤i≤m

X

0≤j≤m j≡i (mod 2)

ε i ε j |s ij |



= 1 4 . P r o o f. This can be proved along the same lines as Corollary 1 in §2 of [20].

R e m a r k. Another proof of Theorem 1.1 which uses a completely dif- ferent method is to be found in [1]. It yields more precise information on where the maximum is attained at the cost of a much longer proof.

2. The maximum order for numbers of bounded density. By a well known theorem of W. M. Schmidt [16] for every α an infinity of positive integers N such that D _N (α) ≥ (66 log 4) ⁻¹ log N exist. On the other hand, it was first observed by H. Behnke [4] that D N (α) = O(log N ) if and only if α = [a ₀ , a ₁ , a ₂ , . . .] is of bounded density (i.e. P _m

i=0 a _i+1 = O(m) as

m → ∞). For these numbers we are now able to compute the infimum of all possible implied constants in the estimate D _N (α) = O(log N ).

Theorem 2.1. Let α be a number of bounded density. Then ν(α) := lim sup

N →∞

D N (α) log N

= 1

4 lim sup

m→∞

1 log q _m

 X ^m

i=0

a i+1 − X

0≤i≤m

X

0≤j≤m j≡i (mod 2)

ε i ε j |s ij |

 .

P r o o f. This may be proved as Theorem 1 in §3 of [20].

Theorem 2.1 implies a property of the function ν which was first shown by L. Ramshaw [15]:

Corollary 2.2. Let α, β be two numbers of bounded density. Assume that there exists a matrix ^{a b} _{c d}

∈ GL 2 (Z) such that β = (aα + b)/(cα + d).

Then ν(β) = ν(α).

P r o o f. This follows immediately from Theorem 2.1 and various parts of the proof of Theorem 2 in §3 of [20].

R e m a r k. The analogous map ν ^∗ (α) = lim sup _{N →∞} D _N ^∗ (α)/ log N is studied in [1, 2]. The image of ν ^∗ has the property ν ^∗ (B) = [ν ^∗ ( √

2), ∞).

(Here B denotes the set of all numbers of bounded density.) In the present case we are able to prove [ν( √

2), ∞) ⊆ ν(B) but ν((1 + √

5)/2) < ν( √ 2).

In the case of quadratic irrationalities there is a formula which does not contain any limit processes:

Theorem 2.3. Let α = [0, a 1 , . . . , a e ] where 2 | e and set

η _t = Y

0≤σ<e σ≡t (mod 2)

(−1) ^a

for t ∈ {0, 1}.

Then

ν(α) = 1

4 log(q _e + αq _e−1 )

 _e−1 X

i=0

a _i+1

+ X 1 t=0

(2t − 1) q _e−1

2η _t − q _e − p _e−1 N

 X

0≤i<e i≡t (mod 2)

ε _i (q _i α − p _i )



+ X 1 t=0

(2t − 1) X

0≤i<e i≡t (mod 2)

X

0≤j<e j≡t (mod 2)

ε _i ε _j q _i p _j sgn (i − j)

 ,

where N denotes the norm of the quadratic field Q(α).

P r o o f. Here the same applies as to Theorem 1 of §4 in [20].

N o t e. In view of Corollary 2.2 the assumption on the shape of the con- tinued fraction expansion of α does not exclude any quadratic irrationalities.

Note also that the period e is not assumed to be of minimal length.

We finish the section with two special cases of Theorem 2.3.

Corollary 2.4. Let α = [0, a, b] with a, b ∈ N. Then

ν(α) = 1

4 log(1 + b/α)



a+b− 1

2 · 1 − (−1) ^a

ab + 2(1 − (−1) ^a ) − 1

2 · 1 − (−1) ^b ab + 2(1 − (−1) ^b )

 . Corollary 2.5. Let α = [0, a] with a ∈ N. Then

ν(α) = a 4 log(1/α)

 1 − 1

2 · 1 − (−1) ^a a ² + 4

 .

R e m a r k. Corollary 2.5 was first proved by L. Ramshaw [15].

3. The maximum order for special Hurwitz continued fractions Theorem 3.1. Let t ∈ N and α t = coth(1/t) = [t, 3t, 5t, . . .]. Then as m → ∞,

1≤N <q max

D _N (α _t ) = 1

4 tm ² + tm + 1

16t ((−1) ^t − 1) log m + O(1).

P r o o f. The proof runs analogously to that of Theorem 1 in §5 of [20].

Theorem 3.2. Let t ∈ N. Then lim sup

N →∞

D _N

 coth 1

t

 log log N log N

 ₂

= t 4 , (1)

lim sup

N →∞ D N ( √

e)

 log log N log N

 ₂

= t 4 , (2)

lim sup

N →∞

D N (

√ e ² )

 log log N log N

 ₂

= 2t + 1 4 . (3)

P r o o f. Using the well known continued fraction expansions of the num- bers coth(1/t), √

e and

√

e ² we proceed according to the following scheme. First we calculate estimates

X m i=0

a i+1 ∼ C 1 (t)m ² and

X m i=1

log a i ∼ C 2 (t)m log m as m → ∞. Since

X m i=1

log a _i ≤ log q _m ≤ X m i=1

log(a _i + 1) = X m

i=1

log a _i + O(m)

we have

log q _m ∼ C ₂ (t)m log m.

For each N ∈ N we define m ∈ N via the relation q _m ≤ N < q _m+1 . Since log q m+1 ∼ log q m as m → ∞ we infer log N ∼ C 2 (t)m log m as N → ∞.

This yields log log N ∼ log m and log N ∼ C ₂ (t)m log m ∼ C ₂ (t)m log log N as N → ∞. Putting all together we find

S _m ∼ X m i=0

a _i+1 ∼ C ₁ (t)m ² ∼ C ₁ (t) C ₂ (t) ²

 log N log log N

 ₂ , where we made use of

X

0≤i≤m

X

0≤j≤m j≡i (mod 2)

ε _i ε _j |s _ij | = O(m).

The result now follows from Corollary 1.2.

4. The minimum order. We continue by introducing a few more no- tations. Let q m ≤ N < q m+1 . There is a unique expansion N = P _m

j=0 b j q j

where 0 ≤ b _j ≤ a _j+1 for all j, b ₀ < a ₁ and b _j = a _j+1 ⇒ b _j−1 = 0 for j ≥ 1.

For j ≥ −1 we define A _j = P _m

µ=0 b _µ s _µj . Let i _N be the smallest integer j ≥ 0 such that b j 6= 0. Set

s := min{j | 2 - j, 1 ≤ j ≤ m, A _j > 0, A _j+2 > 0 ⇒ b _j+1 < a _j+2 } and

t := min{j | 2 - j, 1 ≤ j ≤ m, A _j−1 < 0 < A _j+1 ,

A j+2 > 0 ⇒ b j+1 < a j+2 − 1}, where min ∅ := ∞. Finally, we define

u :=

 0 if 2 | i N and (b 0 < a 1 − 1 or A 1 < 0), min{s, t} otherwise.

Theorem 4.1.

ω ⁺ _N (α) = X

u≤j≤m j≡0 (mod 2)

b j (1 − A j ) + X

u≤j≤m A

<0<A

j≡0 (mod 2)

A j − X

u≤j≤m A

≤0<A

j≡0 (mod 2)

A j

(1)

− X

u≤j≤m A

<0 j≡0 (mod 2)

a _j+1 A _j + (δ _u,0 − 1)A _u ,

ω ⁻ _N (α) = ω ⁺ _N (α) + A 0 − X m j=0

b j ((−1) ^j − A j )

(2)

and

(3) D _N (α) = 2ω _N ⁺ (α) + A ₀ − X m j=0

b _j ((−1) ^j − A _j ).

P r o o f. Though not explicitly stated, (1) and (2) are contained in The- orem 1 of §8 in [17]. (Note the slightly different definition of the A _j .) (3) follows immediately from D N (α) = ω ⁺ _N (α) + ω ⁻ _N (α).

Lemma 4.2. Let q _m ≤ N < q _m+1 . Then ω ⁺ _N (α) = max

1≤k≤N (σ ⁻¹ (k) − N {kα}) ≥ q _m |q _m α − p _m | and

ω ⁻ _N (α) = 1 + max

1≤k≤N (N {kα} − σ ⁻¹ (k)) ≥ q _m |q _m α − p _m |,

where σ : {1, . . . , N } → {1, . . . , N } is the unique permutation which satisfies {ασ(i)} < {ασ(i + 1)} for 1 ≤ i < N .

P r o o f. There is a k ₀ (1 ≤ k ₀ ≤ N ) such that σ ⁻¹ (k ₀ ) = N . We have σ ⁻¹ (k ₀ ) − N {k ₀ α} = N |1 + [k ₀ α] − k ₀ α| ≥ q _m |q _m α − p _m |

as |qα − p| ≥ |q m α − p m | > |q m+1 α − p m+1 | for all (q, p) 6= (q m+1 , p m+1 ) with 0 ≤ q < q _m+1 . The second assumption is proved analogously.

Theorem 4.3. If α is an irrational number , then lim inf

N →∞ D N (α) = 1 + lim inf

m→∞ q m |q m α − p m |.

P r o o f. As in the proof of Corollary 1 in §9 of [17] we compute D bq

(α) (where 1 ≤ b ≤ a _m+1 ) using Theorem 4.1 and arrive at

D _bq

(α) = b − (b − 2)bq _m |q _m α − p _m | − b|q _m α − p _m |.

Putting b = 1 leads to lim inf

N →∞ D N (α) ≤ 1 + lim inf

m→∞ q m |q m α − p m |.

To prove the reverse inequality let ε > 0 and N such that D ^∗ _N (α) = max(ω _N ⁺ (α), ω _N ⁻ (α)) > 1 − ε. (The existence of such an N is guaranteed by Corollary 2 in §9 of [17].) If (without loss of generality) ω _N ⁺ (α) > 1 − ε then D N (α) = ω _N ⁺ (α) + ω _N ⁻ (α) > 1 + q m |q m α − p m | − ε by Lemma 4.2.

R e m a r k. As proved in the above theorem, D _N (α) behaves like D ^∗ _N (α) if the sequence (a _j ) _j≥1 of partial quotients is unbounded, otherwise it is closely related to the Lagrange spectrum. As this set has been studied thoroughly there is an abundance of information available on the set S :=

{lim inf N →∞ D N (α) | α ∈ R\Q} (see [6]). We restrict ourselves to state just

a few of the known facts:

S is a closed subset of the interval [1, 1 + 1/ √

5] with min S = 1 and max S = 1 + 1/ √

5. Its subset S ∩ (1 + 1/3, 1 + 1/ √

5] consists of the numbers 1 + m/ √

9m ² − 4 where m is a positive integer such that m ² + m ² ₁ + m ² ₂ = 3mm ₁ m ₂

for some positive integers m ₁ ≤ m and m ₂ ≤ m. The three largest numbers of S are 1 + 1/ √

5, 1 + 1/ √

8 and 1 + 5/ √

221. Let µ ₀ = 253589820 + 283748 √

462

491993569 = 4.527829 . . .

Then [1, 1+1/µ 0 ] ⊆ S and there is no interval I such that [1, 1+1/µ 0 ] $ I ⊆ S. On the other hand, there are gaps in S such as J = (1+1/ √

13, 1+1/ √ 12), i.e. J ∩ S = ∅ but 1 + 1/ √

12 ∈ S and 1 + 1/ √

13 ∈ S.

References

[1] C. B a x a, Die maximale Gr¨oßenordnung der Diskrepanz der Folge (nα)

, Disser- tation, Universit¨at Wien, 1993.

[2] —, On the discrepancy of the sequence (nα), to appear.

[3] H. B e h n k e, ¨ Uber die Verteilung von Irrationalit¨aten mod 1, Abh. Math. Sem. Univ.

Hamburg 1 (1922), 252–267.

[4] —, Zur Theorie der diophantischen Approximationen I , ibid. 3 (1924), 261–318.

[5] P. B o h l, ¨ Uber ein in der Theorie der s¨akularen St¨orungen vorkommendes Problem, J. Reine Angew. Math. 135 (1909), 189–283.

[6] T. W. C u s i c k and M. E. F l a h i v e, The Markoff and Lagrange Spectra, Math.

Surveys Monographs 30, Amer. Math. Soc., Providence, Rhode Island, 1989.

[7] Y. D u p a i n, R´epartition et discr´epance, Th`ese, Universit´e de Bordeaux I, 1978.

[8] —, Discr´epance de la suite ({n

√5

2

}), Ann. Inst. Fourier (Grenoble) 29 (1979), 81–106.

[9] Y. D u p a i n and V. T. S ´o s, On the discrepancy of (nα) sequences, in: Topics in Classical Number Theory, Vol. 1, Colloq. Math. Soc. J´anos Bolyai 34, G. Hal´asz (ed.), North-Holland, Amsterdam, 1984, 355–387.

[10] G. H. H a r d y and J. E. L i t t l e w o o d, Some problems of Diophantine Approxi- mation: The lattice points of a right-angled triangle II , Abh. Math. Sem. Univ.

Hamburg 1 (1922), 212–249.

[11] E. H e c k e, ¨ Uber analytische Funktionen und die Verteilung von Zahlen mod. Eins, ibid. 54–76.

[12] J. L e s c a, Sur la r´epartition modulo 1 de la suite nα, Acta Arith. 20 (1972), 345–352.

[13] H. N i e d e r r e i t e r, Application of diophantine approximation to numerical integra- tion, in: Diophantine Approximation and Its Applications, C. F. Osgood (ed.), Academic Press, New York, 1973, 129–199.

[14] A. O s t r o w s k i, Bemerkungen zur Theorie der Diophantischen Approximationen, Abh. Math. Sem. Univ. Hamburg 1 (1922), 77–98.

[15] L. R a m s h a w, On the discrepancy of the sequence formed by the multiples of an irrational number, J. Number Theory 13 (1981), 138–175.

[16] W. M. S c h m i d t, Irregularities of distribution VII , Acta Arith. 21 (1972), 45–50.

[17] J. S c h o i ß e n g e i e r, On the discrepancy of (nα), ibid. 44 (1984), 241–279.

[18] J. S c h o i ß e n g e i e r, On the discrepancy of (nα) II , J. Number Theory 24 (1986), 54–64.

[19] —, Absch¨atzungen f¨ ur P

n≤N

B

(nα), Monatsh. Math. 102 (1986), 59–77.

[20] —, The discrepancy of (nα)

, Math. Ann. 296 (1993), 529–545.

[21] W. S i e r p i ń s k i, Sur la valeur asymptotique d’une certaine somme, Bull. Int. Acad.

Polon. Sci. (Cracovie) A (1910), 9–11.

[22] —, On the asymptotic value of a certain sum, Rozprawy Akademii Umiejętności w Krakowie, Wydział mat. przyrod. 50 (1910), 1–10 (in Polish).

[23] V. T. S ´o s, On the theory of diophantine approximation II (inhomogeneous prob- lems), Acta Math. Acad. Sci. Hungar. 9 (1958), 229–241.

[24] —, On strong irregularities of the distribution of {nα} sequences, in: Studies in Pure Mathematics, P. Erd˝os (ed.), Birkh¨auser, Boston, 1983, 685–700.

[25] H. W e y l, ¨ Uber die Gibbs’sche Erscheinung und verwandte Konvergenzph¨ anomene, Rend. Circ. Mat. Palermo 30 (1910), 377–407.

[26] —, ¨ Uber die Gleichverteilung von Zahlen mod. Eins, Math. Ann. 77 (1916), 313–352.

INSTITUT F ¨UR MATHEMATIK UNIVERSIT ¨AT WIEN

STRUDLHOFGASSE 4 A-1090 WIEN, AUSTRIA

E-mail: BAXA@PAP.UNIVIE.AC.AT

Received on 17.12.1993 (2547)