Суммы характеров на сдвинутых степенях Текст научной статьи по специальности «Математика»

ЧЕБЫШЕВСКИЙ СБОРНИК

Том 18 Выпуск 2

УДК 511.321 Б01 10.22405/2226-8383-2017-18-2-267-274

СУММЫ ХАРАКТЕРОВ НА СДВИНУТЫХ СТЕПЕНЯХ1

Ю. Н. Штейников (г. Москва)

Аннотация

Мы изучаем суммы характеров на множестве сдвинутых степеней по модулю простого числа р. Такие суммы могут рассматриваться как обобщение сумм характеров от сдвинутой подгруппы. Случай, когда подгруппа имеет размер меньше /р, вопрос о нетривиальных по порядку верхних оценок таких сумм остается открытым и на сегодня является нерешенным. Он был предложен Ж. Бургейном и М. Ч. Чанг в обзоре 2010 года. Тем не менее, некоторых промежуточных результатов добился профессор К. Гонг, установивший нетривиальные оценки таких сумм в случае когда подгруппа имеет размер существенно больше /р. В данной работе получены некоторые новые результаты на верхнюю оценку абсолютного значения обобщения таких сумм, которые являются неполными суммами характеров от сдвинутых подгрупп. Дано два доказательства основного утверждения. Первое из них основано на сведении указанной суммы к известной оценке А. Вейля и приеме сглаживания сумм. Применяется также прием оценки неполной суммы через полную. Используется также один результат М. 3. Гараева. Второе доказательство основано на оригинальной идее И. М. Виноградова. Этот подход был предложен для уточнения известного неравенства Пойа-Виноградова и использует в своей сути некоторые геометрические и комбинаторные идеи. Второе доказательство приведено не в полной мере. Мы лишь доказываем некоторое ключевое утверждение и за остальными выкладками отсылаем читателя к самой работе И. М. Виноградова.

Ключевые слова: конечные поле, степени, суммы.

Библиография: 15 названий.

CHARACTER SUMS OVER SHIFTED POWERS

Yu. N. Shteinikov (Moscow) Abstract

We study character sums over shifted powers modulo a prime p. Such sums can be viewed as generalizations of character sums over shifted multiplicative subgroups. We obtain some new results on upper estimates for absolute value of these sums. The case when the cardinality of subgroup is less than fp, it is a question of non-trivial upper bounds for such sums that remains open and is unsolved today. It was proposed by J. Burgain and M. Ch. Chang in the review of 2010. Nevertheless, some intermediate results were achieved by Professor K. Gong, who established non-trivial estimates of such sums in the case when the subgroup is much larger than fp. In this paper, we obtain some new results on the upper bound for the absolute value of the generalization of such sums, which are incomplete sums of character sums over shifted subgroups. Two proofs of the main result are given. The first one is based on reduction of this sum to the well-known estimate of A. Weil and the method of smoothing such sums. The method of estimating the incomplete sum through the full one is also applied. One result of M. Z. Garaev is also used. The second proof is based on the original idea of I. M. Vinogradov. This approach was proposed to refine the known inequality of Poya-Vinogradov and uses in its

1 Исследование выполнено за счет гранта Российского научного фонда (проект № 14-11-00433).

essence some geometric and combinatorial ideas. The second proof is not fully presented. We only prove a key statement, and for the rest of the calculations we refer the reader to the initial work of I. M. Vinogradov.

Keywords: finite field, powers, sums.

Bibliography: 15 titles.

1. Introduction

Throughout the paper n - is positive integer and p— is an odd large prime number, Zn be the n— element residue ring and multiplicative group of Zra. Let x '1S some multiplicative character modulo n, gcd(a, n) = 1, адd I £ Z*n, with ord(l) = L, that means L is the minimal positive integer n that ln = 1 (mod n). For 1 < К < L

denote

5(x,a,l,K):= ^ x(a + lk), (a,n) = 1.

1 <k<K

Hong Bing Yu obtained [1] non-trivial upper estimates for S(x, a, l, К). We formulate his result below.

Theorem 1. Let n > 2 and x~ ^ a primitive Dirichlet character modulo n. The following estimate holds

2 7

\S(x,a,l,K)| < v^(-logn

■к 5

Let G с Z^- multiplicative subgroup and

S(x,a,G) :=J2 %(a + x).

xeG

It is an opened question to obtain nontrivial upper estimates for \S(x, a, G)\ when G С Z* and

G ~ p2. Jean Bourgain posed this problem (see M.-C. Chang's 2010 survey). The exact formulation of it is the following.

Problem (J. Bourgain) Let G - subgroup, G С Z*, \G\ ~ p2 ,p ^ ж, x - nontrivial character modulo p, gcd(a,p) = 1. Obtain nontrivial upper estimate in the following form

\S(x,a,G)\ = o(\G\),p ^ ж.

In this paper we consider upper esimates for \S(x, a, l, К)\ in the case of prime modulo p. (x -is a multiplicative character modulo p.)

The main aim of this paper is to give a nontrivial bound for absolute value of S(x, a, l, К) when К is sufficiently large than p2. The main result is contained in the following theorem.

Theorem 2. Let К > p2, x~ is & multiplicative character modulo p. Than the following estimate holds

\S(x,a,l,K)\ p(\og + 1).

In the case of prime modulo the quantities S(x, a, l, К) can be considered as generalizations of S(x, a, G). Indeed, аду subgroup G С Z* is cyclic, that means G = {lk}i<fc<|G| for some I £ Z*. Estimates of character sums for different variants were considered in numerous papers, let us note the works [6],[3], [9],[10], [11],[12], [7], [8].

We recall that the notations A(n) = 0(B(n)),A(n) « В(n) are equivalent to the statement that there exists an absolute constant с that inequalitv \A(n)\ < cB(n) holds for all n.

2. Preliminary results

In this section we collect some facts and lemmas which we will use in the proof of main results. But we shall start with upper estimates for |S(x,a,G)I which were obtained by Ke Gong.

Proposition 1. Let G c Z* -is any multiplicative subgroup, than we have

IS(x,a,G)I< pl/2.

In the proof we use I. M. Vinogradov's lemma, (see it in [13]) which is given below. Lemma 1. Let

S := E E a(x)/3(y)x(xy + a),

0<x<p-l0<y<p-l

with

Then we have

E !«(^)!2E \a(y)\2<y

0<x<p-1 0<у<р-1

|S| < (pXY) 2.

Now we are ready to prove proposition 3. Доказательство. Let G(x) be the indicator of G. We have

Ex(x + a) = |gG E x(xy + a) = ]gG E E G(x)G(y)X(xy + a).

xeG ! ' x,yeG ! ' 0<x<p-10<y<p-1

Applying Vinogradov's lemma to the last sum we obtain the desired estimate. Thus, proposition 3 is proved. □

Let function ^ C. We recall tha Fourier transform f (£) is defined as

1 L

f(.0 := \ E f(x)?-2жг^,

X=1

and

f (x) = E № ^ ^. ?=i

We will use the well known Weil estimate of character sums.

Lemma 2. Let x~ ^ a multiplicative character of Z* of order m > 1 and f <E Zp[x] - is a polynomial with positive degree and with d distinct zeros in Zp and which is not a m-th power of another polynomial. Then we have following estimate

! E X(f(x))! < dP2.

X^&p

The next result belongs to M. Garaev[2] and we will use it.

Lemma 3. Let L1 ,L2, А, В and L be any integers, 1 < А, В < L. Then

L-1 Li+A L2+B

W := El E e2*iax/L\\ ^ e2™y/L\ < la \og(BA-1 + 2)

a=0 x=Ll+1 y=L2+1

Let us denote

f (k) :— X(a + Ik). In the next lemma we get estimates on f(().

Lemma 4. For any £ the following estimate holds

\f (0\<^.

Доказательство. We have f (£) = k=i x(a + lk)e . Let d := Ej-. For some primitive root g modulo p we have I = gd, and we have

1

p — 1

1 p~i

f (0 = —7 E + 9dk Y~2m f.

k=l

Denoting x — g and so k — indg x, we get

p-i

~ 1 v-^ t ^ .^(indgx)

f (0 — ^X(a + xd)e-2™-z8-.

^ x=l

2 .£(indgx)

We see that the function e- z as the function of x is some multiplicative character (x)-Therefore

1 p-1

f(t) = —[Ex(a + xd)Xt (x). P — 1 —i

x=l

Each of the characters is a power of some fixed multiplicative character xl,

X — X?1 ,xs —x?2.

so

1 p-1

m — —TE*i((a+xd)mix?2).

p—1 —i

x=l

Next, we use Weil estimate, - lemma 5 and get the desired estimate. WTith that we finish the proof. □

3. The proof of the main results.

Доказательство. For the proof of theorem 2 we need to estimate | ^n=+A+1 f (n)\> we can assume that sjp < К < p0-99. Let us denote

п+Ш-i 9(n) := E f(k).

k=n

Consider the following sum

A+K

a := E 9(n).

n=A-[^p]+2

It is easy to see that

A+ K

a = Ш( E f(n))+°(P).

n= A+i

So we will get upper estimate for !<т!.

We define I := [A — [^p] +2, A + К]. We have

L-1 L—1f L-1 L-1

- = £9(k) = £ g(k) £ L £ ^ = L £E g(k)e2m^)(£ ^) =

kel k=0 k'el a=0 a=0 k=0 k'el

L-1

= £ 9(a) £ e2ni*

a=0 k'el

For g(a) we have -

l-1[VP\-1 , L-1 [VP]-1

9(a) = L £ £ f(x + k)с-2™" = L £ f(x)( £ ^^) =

x=0 k=0 x=0 k=0

Ш-1

= f(a) £ k=0

So, inserting this to the expression for a we obtain

2-wi

e l .

L-1 Ш-1 ,, ,

- := £ Ka) £ e2™^ £ "

a=0 k''=0 k'el

Recalling estimates for |f(a)I in lemma 7

r L-l [Vp]-1 .. .

ia|< Y- £ i £ e2ni^ ii £ e2ni^ |.

a=0 k''=0 k'el

a

finish the proof of theorem 2. □

4. Final Remarks

There is another way to prove theorem 2.

Professor Ke Gong pointed me on it so I include it here.

It can be done using approach of I. M. Vinogradov [10], where he considered upper bounds for is m<k<M+qX(k)I- We are only going give a sketch of this proof. Denoting

f(k) :=x(a + lk),

so we need to obtain upper bound for | ^^k<K f(k)I. Following the proof as in [10], we need to establish the following key lemma.

Lemma 5. Let P,Ml,..., Mm e N, P < § and

Ml +P <M2,..., Mm-l +P < Mm <Ml + L

Then we have

E

1< i<m

£

£ !(x + z)

Mi+1<x<Mi+P 0<z<P-1

^ (mp)2 P.

Let W is the following set

W :— {x e ZP : 3ie [l,m], 3k e [Mi + 1,Mi + P],x = lk (mod p)} W( x) W

E

l<i<m

E £ f(x+z)

Mi+\<x<Mi+P 0<z<P-1

< e w (x)

0<x<p-1

E X(a + xlz)

0<z<P-1

Next, we use Cauchv-Shwartz inequality and after we should obtain estimate for

2

E

E X(a + xlz)

0<x<p-\ 0<z<P-l

Taking the square of inner sum and changing the order of summation one can easily get the desired estimate. This is the way of proving lemma 8.

The final arguments for the proof of theorem 2 can be found in [10]. There are some kind of geometrical and approximation argumets in the proof.

Acknowledgements

The work is supported by the Russian Science Foundation under grant 14-11-00433. I want to thank professor Jia Chaohua for invition on the Chinese-Russian workshop on exponential sums and sumsets in Beijing in november 2015, for its hospitality and excellent working conditions. The task of this paper appeared while being on that workshop. I also want to thank professors Ke Gong and Sergei Konvagin for valuable advices.

СПИСОК ЦИТИРОВАННОЙ ЛИТЕРАТУРЫ

1. Hong Bing Yu, Estimates of character sums with exponential function // Acta Arithmetica. 2001. Vol. 97, №3, R 211-218.

2. M. Z. Garaev, On the logarithmic factor in error term estimates in certain additive congruence problems 11 Acta Arith. 2006. Vol 124 , P. 27-39.

3. L. Goldmakher, Multiplicative mimicry and improvements of the P?lya-Vino gradov inequality// Algebra and Number Theory 2012. Vol. 6, No. 1, P. 123-163.

Monatsh. Math. 2013. Vol 169, P 317-328.

5. K. Gong, C. Jia, M.A. Korolev, Shifted character sums with multiplicative coefficients, II // J. Number Theory, 2017, Vol 178, P.31-39.

6. D.A. Frolenkov A numerically explicit version of the Poly a-Vinogradov inequality // Moscow journal of Combinatorics and Number Theory, 2011, Vol 1, № 3, P. 25-41.

7. D. A. Frolenkov, K. Soundararajan A generalization of the P о fly a Vinogradov inequality // Ramanujan J., 2013 Vol 31, № 3 , P. 271-279.

8. C. Pomerance Remarks on the Poly a-Vinogradov inequality / / IMRN, 2011, Vol 151, P. 30-41.

9. E. Dobrowolski, K. S. Williams An upper bound for the sum f(n)for a certain class of

funktions/// Proceedings of the American Mathematical Society, 1992, Vol 114, № 1, P. 29-35.

10. I. M. Vinogradov A new improvement of the method of estimation of double sums (Russian)// Dokladv Akad. Nauk SSSR (N.S), 1950, Vol 73, P. 635-638.

11. A. Granville, K. Soundararajan Large character sums: pretentious characters and the Polya-Vinogradov theorem j j J. Amer. Math. Soc. (N.S), 2007, Vol.20, P. 357-384.

12. G. Bachman, L. Rachakonda On a problem of Dobrowolski and William,s and the Polya-Vinogradov inequality j j Ramanujan J., 2001, Vol 5, P. 65-71.

13. I. M. Vinogradov Basics of number theory// Gostechizdat (1952), 1-180.

14. A. A. Karatsuba Basics of analytic number theory. URSS (2004), 1-182.

15. T. Tao , V. Vu Additive combinatorics. Cambridge University Press 2006, P. 1-530.

REFERENCES

1. Hong Bing Yu, Estimates of character sums with exponential function j j Acta Arithmetica 2001. vol. 97, no. 3, pp. 211-218.

2. M. Z. Garaev, On the logarithmic factor in error term estimates in certain additive congruence problems Acta Arith 2006. Vol 124, pp. 27-39.

3. L. Goldmakher, Multiplicative mimicry and improvements of the P?lya-Vino gradov inequality Algebra and Number Theory 2012. vol. 6, no. 1, pp. 123-163.

4. C. Dartvge and A. Sârkôzv, On additive decompositions of the set of primitive roots modulo p Monatsh. Math. 2013. vol 169, pp. 317-328.

5. K. Gong, C. Jia, M.A. Korolev, Shifted character sums with multiplicative coefficients, II J. Number Theory 2017, vol 178, pp. 31-39.

6. D.A. Frolenkov A numerically explicit version of the Poly a-Vinogradov inequality Moscow journal of Combinatorics and Number Theory 2011, vol 1, no. 3, pp. 25-41.

7. D. A. Frolenkov, K. Soundararajan A generalization of the Poflya Vinogradov inequality Ramanujan J. 2013. vol 31, no 3 , pp. 271-279.

8. C. Pomerance Rem,arks on the Poly a-Vinogradov inequality IMRN 2011, vol 151, pp. 30-41.

9. E. Dobrowolski K. S. Williams An upper bound for the sum Yln=a+i f(n)for a certain class of funktions f Proceedings of the American Mathematical Society, 1992, vol 114, no 1, pp. 29-35.

10. I. M. Vinogradov A new improvement of the method of estimation of double sum,s (Russian) Doklady Akad. Nauk SSSR (N.S) 1950, vol 73, pp. 635-638.

11. A. Granville, K. Soundararajan Large character sums: pretentious characters and the Polya-Vinogradov theorem J. Amer. Math. Soc. (N.S) 2007, vol.20, pp. 357-384.

12. G. Bachman, L. Rachakonda On a problem of Dobrowolski and William,s and the Polya-Vinogradov inequality Ramanujan J. 2001, vol 5, pp. 65-71.

13. I. M. Vinogradov Basics of number theory// Gostechizdat (1952), pp. 1-180.

14. A. A. Karatsuba Basics of analytic number theory. URSS (2004), pp. 1-182.

15. Т. Tao , V. Vu Additive combinatorics. Cambridge University Press 2006, pp. 1-530. Математический институт имени В. А. Стеклова

ФГУ ФНЦ Научно-исследовательский институт системных исследований Российской академии наук

Получено 17.03.2017 г. Принято в печать 12.06.2017 г.