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(1)

LXI.3 (1992)

2

by

n≤x

n≤x

2

n≤x

n≤x

2

n≤x

k

2

2

x→∞

n≤x

2

n≤x

k

j

k

j

2

2

2

k

2

k

k

2

2

(2)

k

2

2

2

2

2k

k

2

21

22

2k

2

2k

2

2

jν

j

j

j

(ν)

j

jνj

(ν)

2k

k

(ν)

(ν)

j

j

[2j−1/j]

j

j

j+1

j−1

j

k

[2j/(j+1)]

j+2

2k

2l

k

l

k

l

j

j0

n≤x

a2k+a2l≤x ak≤al

a2k≤x/2

2

2

n≤x

2

2

j

2k

2l

k

l

k

l

j0

(3)

n≤x

2

j≤[log4x]

n

2j

1/2

2

n≤x

2

4

n≤x

2

j≤(log2x)/2

n∈Ij0

2πin2x

4 4

j≤(log2x)/2

n∈Ij0

2πin2x

2

1/2

4 4

j≤(log2x)/2

n∈Ij0

2πin2x

2 4

2

j≤(log2x)/2

n

j2

1/2

2

1

2

1

2

n

j2

j

2

n≤x

2

j≤(log2x)/2

j

2

2

j

2j

1

1

2

2

1

1

2

2

n

2j

rj(n)=1

j

rj(n)≥2

2j

j

j

2

(4)

1

1

2

2

j

2

21

21

22

22

j

j

2

1

1

2

2

1

1

2

2

j

21

21

22

22

j

j

21

21

2

2

2

2

2

2

2

j

21

21

rj(n)≥2

2j

a1,b1

0<|a1/b1|<1/j

j

21

21

2

b1<2j

j

1

3

j

2

(5)

2

2

2

2

1

1

2

2

21

21

22

22

1

1

2

2

2r

2r

2s

2s

j

r

r

j

j

j

s

s

j

j

r

r

s

s

1

1

2

2

21

21

22

22

1

1

r

r

s

s

2

2

r

r

s

s

r

r

s

s

2

2

1

1

2r

2r

2s

2s

s

s

r

r

1

1

2

2

rj(n)2

21

21

22

22

1

1

2

2

j

2j

rj(n)2

2k

2k

2

2

(6)

2

N ≤n≤N +N/ log N

2πin2x

4

1/2

1/2

N ≤n≤N +N/M

2πin2x

4

1/4

1/2

1/2

1/4

### for every M ≥ 1.

References

[1] A. O. L. A t k i n, On pseudo-squares, Proc. London Math. Soc. (3) 14A (1965), 22–27.

[2] J. C i l l e r u e l o and A. C ´o r d o b a, Trigonometric polynomials and lattice points, Proc.

Amer. Math. Soc., to appear.

[3] J. C i l l e r u e l o, B2-sequences whose terms are squares, Acta Arith. 55 (1990), 261–

265.

[4] P. E r d ˝o s and A. R ´en y i, Additive properties of random sequences of positive integers, Acta Arith. 6 (1960), 83–110.

[5] H. H a l b e r s t a m and K. F. R o t h, Sequences, Clarendon Press, Oxford 1966.

and in revised form on 5.7.1991 (2087)

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