Exponential sums involving q-digital functions

2019 ◽  
Vol 15 (06) ◽  
pp. 1143-1172
Author(s):  
Karam Aloui
Keyword(s):  
Real Number ◽  
Exponential Sums ◽  
Exponential Sum ◽  
Similar Estimate ◽  
Short Intervals ◽  
Sum Formula

We estimate the exponential sum [Formula: see text], where [Formula: see text] is a real number and [Formula: see text] are digital functions; in the spirit of the works of Kim and Berend–Kolesnik. A similar estimate along short intervals is also provided.

MORE ON THE SUM-PRODUCT PHENOMENON IN PRIME FIELDS AND ITS APPLICATIONS

2005 ◽  
Vol 01 (01) ◽  
pp. 1-32 ◽  
Author(s):  
J. BOURGAIN
Keyword(s):  
Finite Fields ◽  
Exponential Sums ◽  
Rational Functions ◽  
Exponential Sum ◽  
Kloosterman Sums ◽  
Product Theorem ◽  
Short Intervals ◽  
Prime Fields ◽  
Product Sets ◽  
Mod P

In this paper we establish new estimates on sum-product sets and certain exponential sums in finite fields of prime order. Our first result is an extension of the sum-product theorem from [8] when sets of different sizes are involed. It is shown that if [Formula: see text] and pε < |B|, |C| < |A| < p1-ε, then |A + B| + |A · C| > pδ (ε)|A|. Next we exploit the Szemerédi–Trotter theorem in finite fields (also obtained in [8]) to derive several new facts on expanders and extractors. It is shown for instance that the function f(x,y) = x(x+y) from [Formula: see text] to [Formula: see text] satisfies |F(A,B)| > pβ for some β = β (α) > α whenever [Formula: see text] and |A| ~ |B|~ pα, 0 < α < 1. The exponential sum ∑x∈ A,y∈Bεp(axy+bx2y2), ab ≠ 0 ( mod p), may be estimated nontrivially for arbitrary sets [Formula: see text] satisfying |A|, |B| > pρ where ρ < 1/2 is some constant. From this, one obtains an explicit 2-source extractor (with exponential uniform distribution) if both sources have entropy ratio at last ρ. No such examples when ρ < 1/2 seemed known. These questions were largely motivated by recent works on pseudo-randomness such as [2] and [3]. Finally it is shown that if pε < |A| < p1-ε, then always |A + A|+|A-1 + A-1| > pδ(ε)|A|. This is the finite fields version of a problem considered in [11]. If A is an interval, there is a relation to estimates on incomplete Kloosterman sums. In the Appendix, we obtain an apparently new bound on bilinear Kloosterman sums over relatively short intervals (without the restrictions of Karatsuba's result [14]) which is of relevance to problems involving the divisor function (see [1]) and the distribution ( mod p) of certain rational functions on the primes (cf. [12]).

scholarly journals A note on two-term exponential sum and the reciprocal of the quartic Gauss sums

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Wenpeng Zhang ◽  
Xingxing Lv
Keyword(s):  
Exponential Sums ◽  
Exponential Sum ◽  
Gauss Sums ◽  
Power Mean ◽  
Hybrid Power ◽  
Analytic Methods ◽  
Hybrid Power Mean

AbstractThe main purpose of this article is by using the properties of the fourth character modulo a prime p and the analytic methods to study the calculating problem of a certain hybrid power mean involving the two-term exponential sums and the reciprocal of quartic Gauss sums, and to give some interesting calculating formulae of them.

A Multiple Exponential Sum to Modulus p2

1985 ◽  
Vol 28 (4) ◽  
pp. 394-396 ◽  
Author(s):  
D. R. Heath-Brown
Keyword(s):  
Algebraic Geometry ◽  
Exponential Sums ◽  
Exponential Sum ◽  
Total Degree ◽  
Image Position

AbstractFor suitable polynomials f(x) ∊ ℤ[x] in n variables, of total degree d, it is shown thatThis is, formally, a precise analogue of a theorem of Deligne [1] on exponential sums (mod p). However the proof uses no more than elementary algebraic geometry.

On a New Exponential Sum

2001 ◽  
Vol 44 (1) ◽  
pp. 87-92 ◽  
Author(s):  
Daniel Lieman ◽  
Igor Shparlinski
Keyword(s):  
Exponential Sums ◽  
Exponential Sum ◽  
Multiplicative Order

AbstractLet p be prime and let be of multiplicative order t modulo p. We consider exponential sums of the formand prove that for any ε > 0

ON A CERTAIN GENERAL EXPONENTIAL SUM

2005 ◽  
Vol 01 (02) ◽  
pp. 183-192 ◽  
Author(s):  
H. MAIER ◽  
A. SANKARANARAYANAN
Keyword(s):  
Rational Number ◽  
Upper Bound ◽  
Exponential Sum ◽  
Multiplicative Functions ◽  
Sum Formula

In this paper we study the general exponential sum related to multiplicative functions f(n) with |f(n)| ≤ 1, namely we study the sum [Formula: see text] and obtain a non-trivial upper bound when α is a certain type of rational number.

On a variant of sum-product estimates and explicit exponential sum bounds in prime fields

2009 ◽  
Vol 146 (1) ◽  
pp. 1-21 ◽  
Author(s):  
J. BOURGAIN ◽  
M. Z. GARAEV
Keyword(s):  
Prime Order ◽  
Exponential Sums ◽  
Exponential Sum ◽  
Complex Numbers ◽  
Number Of Solutions ◽  
Prime Fields ◽  
Explicit Bounds ◽  
Multiplicative Subgroup ◽  
Image Position

AbstractLet Fp be the field of a prime order p and F*p be its multiplicative subgroup. In this paper we obtain a variant of sum-product estimates which in particular implies the bound for any subset A ⊂ Fp with 1 < |A| < p12/23. Then we apply our estimate to obtain explicit bounds for some exponential sums in Fp. We show that for any subsets X, Y, Z ⊂ F*p and any complex numbers αx, βy, γz with |αx| ≤ 1, |βy| ≤ 1, |γz| ≤ 1, the following bound holds: We apply this bound further to show that if H is a subgroup of F*p with |H| > p1/4, then Finally we show that if g is a generator of F*p then for any M < p the number of solutions of the equation is less than $M^{3-1/24+o(1)}\Bigl(1+(M^2/p)^{1/24}\Bigr).$. This implies that if p1/2 < M < p, then

The Estimation of Complete Exponential Sums

1985 ◽  
Vol 28 (4) ◽  
pp. 440-454 ◽  
Author(s):  
J. H. Loxton ◽  
R. C. Vaughan
Keyword(s):  
Exponential Sums ◽  
Exponential Sum ◽  
Integer Coefficients ◽  
Complex Zeros

AbstractThis paper proves a conjecture of Loxton and Smith about the size of the exponential sum S(f;q) formed by summing exp (2πif(x)/q) over x mod q, where f is a polynomial of degree n with integer coefficients. It is shown that |S(f;q)| ≤ Cfdn(q)qe/(e+1), where e is the maximum of the orders of the complex zeros of f'. An estimate is also obtained for Cf in terms of n, e and the different of f, and a number of examples are given to show that the estimate is best possible.

scholarly journals The mean value of the generalized Dirichlet L-functions with the weight of the Gauss sums

2021 ◽  
pp. 1-18
Author(s):  
Yana Niu ◽  
Rong Ma ◽  
Yulong Zhang ◽  
Peilin Jiang
Keyword(s):  
Asymptotic Formula ◽  
Real Number ◽  
Analytic Continuation ◽  
Dirichlet Character ◽  
Mean Value ◽  
Gauss Sums ◽  
Number Formula ◽  
Sum Formula ◽  
The Mean ◽  
Generalized Dirichlet

Let [Formula: see text] be an integer, and let [Formula: see text] denote a Dirichlet character modulo [Formula: see text]. For any real number [Formula: see text], we define the generalized Dirichlet [Formula: see text]-function as [Formula: see text] where [Formula: see text] with [Formula: see text] and [Formula: see text] both real. It can be extended to all [Formula: see text] using analytic continuation. For any integer [Formula: see text], the famous Gauss sum [Formula: see text] is defined as [Formula: see text] where [Formula: see text]. This paper uses analytic methods to study the mean value properties of the generalized Dirichlet [Formula: see text]-functions with the weight of the Gauss sums, and a sharp asymptotic formula is obtained.

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