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# Kloosterman-type sums and the discrepancy of nonoverlapping pairs of inversive congruential pseudorandom numbers

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(1)

ω

n

n

n

m

n

n≥0

m

n+1

−1n

n−1

n

n

n≥0

n

n

0

1

m/2−1

m

n

n≥0

n

2n

2n+1

0

1

m/4−1

2

m/4(2)

(2)

m/4(2)

R

m/4

2

m/4

0

1

m/4−1

m/4(2)

(2)m/4

(2)m/4

3/2

2

−1/2

2

−1/2

−1/2

−1

(2)m/4

m/4(2)

3/2

−1/2

(2)m/4

−1/2

2

0

(2)m/4

−1/2

2

−1/2

1/2

k

1

k

k

j

(3)

1

k j=1

j

1

k

k

2πit

k

(2)m/4

(2)m/4

h∈C2(m)

m/4−1

n=0

n

(2)m/4

m/4(2)

1

2

m/4−1

n=0

n

1

2

2

1

2

8

## . Then X

k∈C1(2t) k≡c (mod 8)

t

t

m/4−1

n=0

n

m/4(2)

0

2

m/2−2

m

0

m/4−1

n=0

n

m/4−1

n=0

1

2n

2

2n+1

m/4−1

n=0

1

2n

2

−12n

y∈Zm

y≡y0(mod 4)

1

2

−1

(4)

1

2

2

α

y∈Z y≡ξ (mod 4)

−1

α

1

2

2

2

α

22α

α

α

α

2α−1

2α

α

α

α

22α

α

α

α

0

0

0

0

α−1

α−1

α

α−1

α−1

α

3

3

3

α+1

α

α

α

α

22α

α

α

α

α

α

α

α

α

α+1

2

22α+1

α

α

α+1

α

α

α−1

α

2

α

α

α−1

α

α+1

(5)

2

α

α−1

α

α

α−1

α

α

α

α

α

α+1

22α+1

α+1

2

α

α−1

α

α−1

α

0

0

α+1

0

2

α+1

α

α

α−1

α

(α+1)/2

α

## = X

y∈Z2α−1 y≡ξ (mod 4)

−1

α

α−1

α−1

−1

α

## = X

y∈Z2α−1 y≡ξ (mod 4)

−1

α

(6)

α

## ) = 2 X

y∈Z2α−1 y≡ξ (mod 4)

−1

α−1

α−1

−1

−1

α

α

η∈Z2α−2

−1

α

η∈Z2α−2

α−2

α−2

α−2

α−2

α

2

α

α

## = X

y,z∈Z y≡z≡ξ (mod 4)

−1

−1

α

y,z∈Z

y≡z≡ξ (mod 4)

1

−1

2

α

1

2

α−1

α−1

α

α

2

(y,z)∈Z2α−1×Z

y≡z≡ξ (mod 4)

1

−1

2

α

(s,t)∈Nα

−1

α

1

2

1

s∈Z s≡1 (mod 8)

t∈Z t≡0 (mod 8)

−1

α

(7)

2

s∈Z s≡5 (mod 8)

t∈Z t≡4 (mod 8)

−1

α

1

s∈Z s≡1 (mod 8)

τ ∈Z2α−3

−1

α−3

α−3

2α

α−3

α

2

s∈Z s≡5 (mod 8)

τ ∈Z

2α−2

−1

α−2

## = 4 X

s∈Z2α−3 s≡5 (mod 8)

τ ∈Z

2α−2

−1

α−2

α−3

−1

α−2

## = 4 X

s∈Z2α−3 s≡5 (mod 8)

τ ∈Z

2α−2

−1

α−2

2α−2

(2)m/4

h∈C2(m)

1

2

0

(2)m/4

## X

h∈C2(m) h≡0 (mod 2ω−2)

1

2

ω−3

γ=0

## X

h∈C2(m) gcd(h1,h2,m)=2γ

1

2

(8)

## m + X

h∈C2(m) h≡0 (mod 2ω−2)

ω−3

γ=0

γ

## X

h∈C2(m) gcd(h1,h2,m)=2γ

1

γ

2

γ

ω−γ

k∈C1(4)

ω−2

2

ω−3

γ=0

γ

## X

k∈C2(2ω−γ) gcd(k1,k2,2)=1

γ

1

2

ω−γ

2

ω−3

γ=0

γ

## X

k∈C2(2ω−γ) k1≡k2≡1 (mod 2)

γ

1

2

ω−γ

2

ω−3

γ=0

−γ

## X

k∈C2(2ω−γ) k1≡k2≡1 (mod 2)

ω−γ

1

2

ω−γ

2

2

## X

k∈C2(8) k1≡k2≡1 (mod 2)

2

## X

k∈C2(16) k1≡k2≡1 (mod 2)

k1≡k2(mod 4)

2

## X

k∈C2(32) k1≡k2≡1 (mod 2) k1≡5k2a (mod 8)

ω−6

γ=0

−γ+(ω−γ+1)/2

## X

k∈C2(2ω−γ) k1≡k2≡1 (mod 2)

k1≡k2a (mod 8)

ω−γ

2

2

##  X

k∈C1(8) k≡1 (mod 2)

2

2

##  X

k∈C1(16) k≡1 (mod 4)

2

(9)

2

d∈{1,3}

##  X

k∈C1(32) k≡d (mod 8)

1/2

ω−6

γ=0

−3γ/2

d∈{1,3}

ω−γ

##  X

k∈C1(2ω−γ) k≡d (mod 8)

ω−γ

(2)m/4

2

2

d∈{1,3}

##  X

k∈C1(32) k≡d (mod 8)

2

1/2

ω−6

γ=0

−3γ/2

d∈{1,3}

##  X

k∈C1(2ω−γ) k≡d (mod 8)

ω−γ

2

2

1/2

ω−6

γ=0

−3γ/2

2

ω−γ

2

ω−γ

1/2

ω−5

γ=0

−3γ/2

2

ω−γ

2

ω−γ

1/2

γ=0

−3γ/2

2

2

3/2

1/2

2

2

(B−1)/2

2

(10)

(2)m/4

(B−1)/2

0

(B−1)/2

(B−1)/2

1/2

## , which completes the proof.

α

### [11] —, Pseudorandom numbers and quasirandom points, Z. Angew. Math. Mech., to appear.

SCHLOSSGARTENSTRASSE 7 DER WISSENSCHAFTEN

F.R.G. A-1010 WIEN

AUSTRIA E-mail: NIED@QIINFO.OEAW.AC.AT

Cytaty

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