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392
result(s) for
"Exponential sums."
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How many Fourier coefficients are needed?
We are looking at families of functions or measures on the torus which are specified by a finite number of parameters
N
. The task, for a given family, is to look at a small number of Fourier coefficients of the object, at a set of locations that is predetermined and may depend only on
N
, and determine the object. We look at (a) the indicator functions of at most
N
intervals of the torus and (b) at sums of at most
N
complex point masses on the multidimensional torus. In the first case we reprove a theorem of Courtney which says that the Fourier coefficients at the locations
0
,
1
,
…
,
N
are sufficient to determine the function (the intervals). In the second case we produce a set of locations of size
O
(
N
log
d
-
1
N
)
which suffices to determine the measure.
Journal Article
The Fourth Hybrid Power Mean Involving the Character Sums and Exponential Sums
In this paper, we consider the fourth hybrid power mean involving two-term exponential sums and third-order character sum modulo p, a topic of significant importance in analytic number theory. These results generalize prior research, and provide new insights for studying the relationship between character sums and exponential sums.
Journal Article
Decoupling, exponential sums and the Riemann zeta function
We establish a new decoupling inequality for curves in the spirit of earlier work of C. Demeter and the author which implies a new mean value theorem for certain exponential sums crucial to the Bombieri-Iwaniec method as developed further in the work of Huxley. In particular, this leads to an improved bound |ζ(12+it)|≪t13/84+ε|\\zeta (\\frac {1}{2} + it)| \\ll t^{13/84 + \\varepsilon } for the zeta function on the critical line.
Journal Article
Hypergeometric functions over finite fields
Building on the developments of many people including Evans, Greene, Katz, McCarthy, Ono, Roberts, and Rodriguez-Villegas, we
consider period functions for hypergeometric type algebraic varieties over finite fields and consequently study hypergeometric functions
over finite fields in a manner that is parallel to that of the classical hypergeometric functions. Using a comparison between the
classical gamma function and its finite field analogue the Gauss sum, we give a systematic way to obtain certain types of hypergeometric
transformation and evaluation formulas over finite fields and interpret them geometrically using a Galois representation perspective. As
an application, we obtain a few finite field analogues of algebraic hypergeometric identities, quadratic and higher transformation
formulas, and evaluation formulas. We further apply these finite field formulas to compute the number of rational points of certain
hypergeometric varieties.
eBook
Theory and applications of finite fields : the 10th International Conference on Finite Fields and Their Applications, July 11-15, 2011, Ghent, Belgium
This volume contains the proceedings of the 10th International Congress on Finite Fields and their Applications (Fq 10), held July 11-15, 2011, in Ghent, Belgium. Research on finite fields and their practical applications continues to flourish. This volume's topics, which include finite geometry, finite semifields, bent functions, polynomial theory, designs, and function fields, show the variety of research in this area and prove the tremendous importance of finite field theory.
eBook
Exponential sums involving automorphic forms for GL(3) over arithmetic progressions
Let f be a Hecke-Maass cusp form for SL(3,ℤ) with Fourier coeffcients Af (m, n), and let ϕ(x) be a C∞-function supported on [1, 2] with derivatives bounded by ϕ(j)(x)≪j 1: We prove an asymptotic formula for the nonlinear \\[\\Sigma_{n\\equiv l \\rm{mod} \\it{q}}\\]Af (m, n)ϕ(n/X)e(3(kn)1/3/q), where e(z) = e2πiz and k ∈ℤ+.
Journal Article
Exponential sums: Questions by Denef, Sperber, and Igusa
We prove the remaining part of the conjecture by Denef and Sperber [Denef, J. and Sperber, S., Exponential sums mod pnp^n and Newton polyhedra, Bull. Belg. Math. Soc., suppl. (2001) 55-63] on nondegenerate local exponential sums modulo pmp^m. We generalize Igusa’s conjecture in the introduction of [Igusa, J., Lectures on forms of higher degree, Lect. Math. Phys., Springer-Verlag, 59 (1978)] from the homogeneous to the quasi-homogeneous case and prove the nondegenerate case as well as the modulo pp case. We generalize some results by Katz in [Katz, N. M., Estimates for “singular” exponential sums, Internat. Math. Res. Notices (1999) no. 16, 875-899] on finite field exponential sums to the quasi-homogeneous case.
Journal Article
Expander estimates for cubes
If A is a set of natural numbers of exponential density , then the exponential density of all numbers of the form x^3+a with xınN and aınA is at least (1, 13+56 ) . This is a considerable improvement on the previous best lower bounds for this problem, obtained by Davenport more than 80 years ago. The result is the best possible for 45 .
Journal Article