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# 1. Introduction. In this paper we extend the results of [6] to pure and mixed exponential sums of the type

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(1)

m

pm

x=1

pm

m

pm

x=1

pm

m

m

pm

pm

m

2πix/pm

1

2

1

2

1

2

2

2

1

2

2

2

m

1

2

1

2

p

1

2

p

p

p

1/2

[67]

(2)

p

p

n

i=1

i

i

i

i

p

1/2

p

1/2

p

1

p

2

p

2

1/2

p

1

p

2

p

1/2

p

1

p

2

p

2

1/2

p

1

p

2

−1

m

m(1−1/d)

p

m

m(1−1/(d+1))

p

ordp(x)

p

(3)

0

1

d

d

p

0≤i≤d

p

i

p

1

2

p

1

p

2

p

0

p

−t

0

m

α

α

2

α

α

m

pm

x≡α (mod p)x=1

pm

α

m

α

t/(να+1)

m(1−1/(να+1))

m

α∈A

α

t/(M +1)

m(1−1/(M +1))

α∈A

α

α

p

k

m

m(1−1/d)

p

α

p

(4)

p

α

m

α

m(1−1/σα)

α

α

α

p−1

m

m

m−1

k

m−1

0

m−1

m−1

1

1

p

0

p

1

1

p

p

m

−t1

0

α

α

α

α

m

pm

x≡α (mod p)x=1

pm

α

m

α

t/(να+1)

m(1−1/(να+1))

m

α∈A

α

t/(M +1)

m(1−1/(M +1))

(5)

α∈A

α

α

α

0

2

10

1

20

22

22

1

2

2

m

m(1−1/(D+1))

p

m

1

1

2

1

2

1

m

m(1−1/(D+1))

m

m

m(1−1/3)

m(1−1/(D+1))

1

2

1

2

−1

pm

x=1

pm

−1

m/2

2

2

2

2

2

2

2

2

2m/3

2

2

2

−1

(6)

m/2

m

2m/3

p

p

p

p

1

2

1

2

p

2

p

p

1

p

2

p

i=0

i

i

i

i

(i)

p

p

0

i=0

i

i−1

p

0

1

1

p

1

ν

p

2

2

2

i

i=1

p

i≥0

p

i

i

ν

t

p

t

p

p

1

p

2

t

p

(7)

k

p

k

k

d

i=1

i

i

1

2

1

2

1

d1

i=0

i

i

2

d2

i=0

i

i

p

i=0

i

i

1

d1

i=0

i

i

2

d2

i=0

i

i

1

2

2

2

p

k

1

k

0≤i≤d

1

p

i

i≥0

p

i

p

1

p

1

2

1

2

p

i=0

i

i

1

d1

i=ν

i

i

2

d2

i=0

i

i

1

2

0

0

i≥1

p

i

t

ν

ν

t

p

t

p

t

p

1

p

1

2

2

p

p

p

t

p

dp

dp

t

p

(8)

−t

0

p

i

p

i

i

p

i

p

p

−σ

1

p

0

1

−τ

0

i=1

i

i−σ

i

1

i=1

i

i−σ−τ

i−1

1

p

p

1

p

p

p

p

p

p

1

τ

p

p

p

i=1

i

i

i

(9)

i≥1

p

i

i

p

p

i

i

p

i

p

p

p

i

p

1

p

ν+1

ν+1−σ−τ

p

1

i=1

i

i

p

ω

p

ω

ω

p

1

2

1

2

p

0

1

1

1

1

1

2

p

m−t1−1

0

m−t1−1

m

2

i=0

i

i

i

p

1

m−t1−1

i=0

i

m−t1−1

i

p

i

p

i

(m−t1−1)i

1

p

1

1

p

(10)

1

p

1

1

1

1

pm

p

p

pm

pm

p

m

m

1

2

2

p

pm

pm

1

2

2

2

m

1

2

1

2

m

2

α

α

m

x≡α (mod p)x=1

pm

2

p

0

α

m

α

m

t/(ν+1)

m(1−1/(ν+1))

α

m

pm

(m+t)/2

2

α

pm

p

(m+t−1)/2

−t

0

[(m−t+1)/2]

α

−t

00

(11)

α

α

m

t/(ν+1)

m(1−1/(ν+1))

1

α

t+1

## X

y≡α (mod p) pt+1|f0(y)

pm

α

α

p

α

−σ

α

m

pm

σ−1

α

m−σ

α

m−σ

m−σ

α

α

m

m

p

m

m

(12)

p

0

α

m

m(1−1/σ)

α

m

m(1−1/σ)

p

m

m(1−1/d)

α

p

m

α∈A

α

m(1−1/dp(f ))

p

1

m(1−1/dp(f ))

1

−t

0

p

1

p

t

m

m

t

m(1−1/d)

t

m

d

d

m/d

m

m

m(1−1/d)

α

m(1−1/σ)

(13)

α

(m+t)/2

α

m−1

m(1−1/m)

m(1−1/σ)

α

m−1

m(1−1/σ)

σ

σ

p

p

α

α

p

σ

p

p

p

p

12

p

p

p

p

p

p

p

α

σ−1

α

p

α

σ−1/2

m(1−1/σ)

1

p

0α

1

−τ

α0

α

m−1

m(1−1/σ)

m−σ

σ

τ +1

σ

τ

p

τ +1

2

2

σ

τ

i=0

i

i

i

(14)

p

3

p

3

p

3

α

2

α

2

α

−3

i=1

i

i

3

1

2

2

α

3

3

4

4

2

2

1

2

1

2

α

α

2

1 y=−1

9

α

α

3

9

9

1/3

α

2

α

2

3

1/3

5(1−1/σ)

α

m−σ

m

σ−1

p

1

## )p

(m−σ)(1−1/dp(gα))

p

1

p

α

m

σ−1

(m−σ)(1−1/σ)

m(1−1/σ)

0

α

m

(m+t)/2

m(1−1/σ)

α

3

3(1−1/σ)

(15)

α

m

m−1

m(1−1/σ)

m−σ

τ +2

σ

σ

τ

p

α

m

α

α

m

m

2

1

1

α

m

1

α

m

t/(ν+1)

m(1−1/(ν+1))

α

m

pm

(m+t)/2

pm

2

α

p

(m+t−1)/2

−t

0

[(m−t+1)/2]

α

−t

0

00

p

p

p−1

x=0

p

2

p−1

x=1

2

p

2

−1

−1

i=1

i+1

i

(16)

m

α∈A

α

t/(M +1)

m(1−1/(M +1))

m

m

−t1

0

2m−2

m−2

1

2

0

1

0

t+1

m

1

m

1

m

t/(ν+1)

m(1−1/(ν+1))

m

1

1

1

2

1

2

1

2

2

p

m

m

m(1−1/(D+1))

1

t

m

m

t

m(1−1/(D+1))

(17)

1

1

α∈A

α

m

t/(D+1)

m(1−1/(D+1))

1/(D+1)

m(1−1/(D+1))

1

m

m−1

t+1

p

1

p

0

1

p

0

1

p

p

i

i=−l

i

i

k

p

i

p

1

2

1

2

1

2

p

2

2

2

l

1

2

1

2

1

1

2

1

l

1

p

1

1

1

1

1

i=0

i

i

i

p

0

## (X) + c) ≥ k

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