Estimates For Exponential Sums Research Articles

Sequences related to Legendre/Jacobi sequences

Two families of binary sequences are constructed from dth order cyclotomic generator and from dth order generalized cyclotomic generator with respect to two distinct primes respectively. By using estimates of certain exponential sums over rings or fields, the upper bounds of both the well-distribution measure and the order k (at least for small k) correlation measure of the resulting binary sequences are evaluated, and some lower bounds on the linear complexity profile of d-ary cyclotomic sequences and d-ary generalized cyclotomic sequences are presented. Our results indicate that such binary sequences possess “very good” local pseudo-randomness.

Numbers with fixed sum of digits in linear recurrent number systems

We study the set of integers with a given sum of digits with respect to a linear recurrent digit system. An asymptotic formula for the number of integers ≤N with given sum of digits is determined, and the distribution in residue classes is investigated, thus generalizing results due to Mauduit and Sarkozy. It turns out that numbers with fixed sum of digits are uniformly distributed in residue classes under some very general conditions. Namely, the underlying linear recurring sequence must have the property that there is no prime factor P of the modulus such that all but finitely many members of the sequence leave the same residue modulo P. The key step in the proof is an estimate for exponential sums using known theorems from Diophantine approximation.

Exponential sum estimates over subgroups and almost subgroups of $$ \mathbb{Z}_{Q}^{*} $$ , where Q is composite with few prime factors

In this paper we extend the exponential sum results from [BK] and [BGK] for prime moduli to composite moduli q involving a bounded number of prime factors. In particular, we obtain nontrivial bounds on the exponential sums associated to multiplicative subgroups H of size qδ, for any given δ > 0. The method consists in first establishing a ‘sumproduct theorem’ for general subsets A of \(\mathbb{Z}^{q} \) . If q is prime, the statement, proven in [BKT], expresses simply that either the sum-set A + A or the product-set A.A is significantly larger than A, unless |A| is near q. For composite q, the presence of nontrivial subrings requires a more complicated dichotomy, which is established here. With this sum-product theorem at hand, the methods from [BGK] may then be adapted to the present context with composite moduli. They rely essentially on harmonic analysis and graph-theoretical results such as Gowers’ quantitative version of the Balog–Szemeredi theorem. As a corollary, we get nontrivial bounds for the ‘Heilbronn-type’ exponential sums when q = p r (p prime) for all r. Only the case r = 2 has been treated earlier in works of Heath-Brown and Heath-Brown and Konyagin (using Stepanov’s method). We also get exponential sum estimates for (possibly incomplete) sums involving exponential functions, as considered for instance in [KS].

Mean values of Dirichlet polynomials and applications to linear equations with prime variables

We prove a new mean-value theorem for Dirichlet polynomials with coefficients given by the von Mangoldt function. We then use our theorem to derive new estimates for certain exponential sums over primes. The latter have applications to additive problems with prime variables. In particular, we are able to improve on a recent result of J.Y. Liu and K.M. Tsang on the size of the solutions of linear equations with prime variables.

Estimation of certain exponential sums arising in complexity theory

It is shown that the correlation on { 0 , 1 } n between parity and a polynomial p ( x 1 , … , x n ) ∈ Z [ X 1 , … , X n ] ( mod q ) , q a fixed odd number and p ( X ) of degree d arbitrary but fixed, is exponentially small in n as n → ∞ . An application to circuit complexity, from where the problem originates, is given. To cite this article: J. Bourgain, C. R. Acad. Sci. Paris, Ser. I 340 (2005).

Mean-value estimates for nonlinear Weyl sums over primes

We establish a Bombieri-type mean-value estimate for nonlinear ex-ponential sums over primes. This has applications in the Waring-Goldbach problem.

Sum-product theorem and exponential sum estimates in residue classes with modulus involving few prime factors

In this Note, we extend the results of Bourgain, Konyagin and Glibichuk to certain composite moduli q involving few ‘large’ primes. First a ‘sum-product’ theorem for subsets A of Z q is obtained, ensuring that | A + A | + | A . A | > c | A | 1 + ɛ provided | A | < q 1 − δ and A does not have a ‘large’ intersection with a translate of a subring. Next, exponential sum estimates are established. In particular nontrivial bounds are obtained for the exponential sums associated to a multiplicative subgroup H < Z q * , with applications to Heilbronn-type sums. To cite this article: J. Bourgain, M.-C. Chang, C. R. Acad. Sci. Paris, Ser. I 339 (2004).

Mordell type exponential sum estimates in fields of prime order

We establish a Mordell type exponential sum estimate (see Mordell [Q. J. Math. 3 (1932) 161–162]) for ‘sparse’ polynomials f ( x ) = ∑ i = 1 r a i x k i , ( a i , p ) = 1 , p prime, under essentially optimal conditions on the exponents 1 ⩽ k i < p − 1 . The method is based on sum–product estimates in finite fields F p and their Cartesian products. We also obtain estimates on incomplete sums of the form ∑ s = 1 t e p ( ∑ i = 1 r a i θ i s ) for t > p ɛ , under appropriate conditions on the θ i ∈ F p * . To cite this article: J. Bourgain, C. R. Acad. Sci. Paris, Ser. I 339 (2004).

On some congruence with application to exponential sums

We will study the solution of a congruence,x ≡g1/2)ωg(2n) mod 2n, depending on the integersg andn, where ωg(2n) denotes the order ofg modulo 2n. Moreover, we introduce an application of the above result to the study of an estimation of exponential sums.

Exponential sums over Mersenne numbers

We give estimates for exponential sums of the form P n6N �( n)exp(2 �iag n /m ), where m is a positive integer, a and g are integers relatively prime to m, andis the von Mangoldt function. In particular, our results yield bounds for exponential sums of the form P p6N exp(2 �iaM p/m ), where Mp is the Mersenne number; Mp = 2 p !1 for any prime p. We also estimate some closely related sums, including P n6N µ(n)exp(2 �iag n /m ) and P n6N µ 2 (n)exp(2 �iag n /m ), where µ is the Mobius function.

0, 1 distribution in the highest level sequences of primitive sequences over

In this paper, we discuss the 0,1 distribution in the highest level sequence a e -1 of primitive sequence over Z 2 e generated by a primitive polynomial of degree n . First we get an estimate of the 0,1 distribution by using the estimates of exponential sums over Galois rings, which is tight for e relatively small to n . We also get an estimate which is suitable for e relatively large to n . Combining the two bounds, we obtain an estimate depending only on n , which shows that the larger n is, the closer to 1/2 the proportion of 1 will be.

Some applications of Bombieri's estimate for exponential sums

Some applications of Bombieri's estimate for exponential sums

Exponential sums along p-adic curves

Let K be a p-adic field, R the valuation ring of K, and P the maximal ideal of R. Let Y⊆ R 2 be a non-singular closed curve, and Y m its image in R/ P m × R/ P m , i.e. the reduction modulo P m of Y. We denote by Ψ an standard additive character on K. In this paper we discuss the estimation of exponential sums of type S m ( z, Ψ, Y, g)≔∑ x∈ Y m Ψ( zg( x)), with z∈ K, and g a polynomial function on Y. We show that if the p-adic absolute value of z is big enough then the complex absolute value of S m ( z, Ψ, Y, g) is O( q m(1− β( f, g)) ), for a positive constant β( f, g) satisfying 0< β( f, g)<1.

On the Uniformity of Distribution of the Elliptic Curve ElGamal Signature

We show that, under some natural conditions, the pairs (ρ, σ) produced by the elliptic curve ElGamal signature scheme are uniformly distributed. In particular, this implies that values of ρ and σ are not correlated. The result is based on some new estimates of exponential sums. For the ElGamal signature over a finite field, a similar result has been obtained by the second author.

On the Uniformity of Distribution of the Elliptic Curve ElGamal Signature

We show that, under some natural conditions, the pairs ( ρ, σ) produced by the elliptic curve ElGamal signature scheme are uniformly distributed. In particular, this implies that values of ρ and σ are not correlated. The result is based on some new estimates of exponential sums. For the ElGamal signature over a finite field, a similar result has been obtained by the second author.

On the Uniformity of Distribution of the ElGamal Signature

We show that, under some natural conditions,the pairs (r,s) produced by the ElGamal signature scheme are uniformly distributed. In particular this implies that values of r and s are not correlated. The result is based on some new estimates of exponential sums.

Double exponential sums over thin sets

We estimate double exponential sums of the form Sa(X, Y) = E E exp(27riaI0xY/p), xGE X yG Y where V is of multiplicative order t modulo the prime p and X and Y are arbitrary subsets of the residue ring modulo t. In the special case t = p 1, our bound is nontrivial for I XI > I Yl > p15/16+6 with any fixed 6 > 0, while if in addition we have Xl > p1-6/4 it is nontrivial for I YI ? p3/4+6. Let p be a prime and let Fp be a finite field of p elements. For an integer m > 1 we denote by Z,m = {0, . .. , m 1} the residue ring modulo m. We also identify Fp with the set {o,... ,p-1}. Finally we define e(z) = exp(27ri/p) and use log z for the natural logarithm of z. Throughout the paper the implied constants in symbols 'O', ' ' may occasionally, where obvious, depend on the small positive parameter E and are absolute otherwise (we recall that A A are equivalent to A 0 (B)). We fix an element ) E IFp of multiplicative order t, that is, 79s 7 1, 1 pl5/16+6 with any fixed 6 > 0. Further examples are given below. Our results rely on the following estimate for certain double exponential sums from [1]; see the proof of Theorem 8 of that paper. Let A E Fp be of multiplicative order T. For any a, b (E IF we have the bound 4 (1) E| e(a\v +bAuv) 1 satisfies (2) T(m) < m(1). 2 Our main estimate is the following. Theorem. For any sets X, Y C Zt, the bound max ISa( X, Y)| << I x 1/21 yI5/6tl/2pl/8+E aEF* p holds. Proof. For a divisor dlt we denote by Y(d) the subset of y Y Y with gcd(y, t) d. Then ISa(X, Y)I ? Z ud| dlt

Euclidean weights of codes from elliptic curves over rings

We construct certain error-correcting codes over finite rings and estimate their parameters. For this purpose, we need to develop some tools, notably an estimate for certain exponential sums and some results on canonical lifts of elliptic curves. These results may be of independent interest.

The CREW PRAM Complexity of Modular Inversion

One of the long-standing open questions in the theory of parallel computation is the parallel complexity of the integer gcd and related problems, such as modular inversion. We present a lower bound $\Omega (\log n)$ for the parallel time on a concurrent-read exclusive-write parallel random access machine (CREW PRAM) computing the inverse modulo certain n-bit integers, including all such primes. For infinitely many moduli, our lower bound matches asymptotically the known upper bound. We obtain a similar lower bound for computing a specified bit in a large power of an integer. Our main tools are certain estimates for exponential sums in finite fields.

On the Representations of a Number as the Sum of Three Cubes and a Fourth or Fifth Power

Let RkOnU denote the number of representations of a natural number n as the sum of three cubes and a kth power. In this paper, we show that R3OnUn 5=9ae ,a nd that R4OnUn 47=90ae ,w heree > 0 is arbitrary.This extends workof Hooley concerning sums of four cubes, to the case of sums of mixed powers. To achieve these bounds, we use a variant of the Selberg sieve method introduced by Hooley to study sums of two kth powers, and we also use various exponential sum estimates.