Estimates For Exponential Sums Research Articles

On Artin’s Primitive Root Conjecture for Function Fields over 𝔽q

ABSTRACT In 1927, E. Artin proposed a conjecture for the natural density of primes p for which g generates $(\mathbb{Z}/p\mathbb{Z})^\times$. By carefully observing numerical deviations from Artin’s originally predicted asymptotic, Derrick and Emma Lehmer (1957) identified the need for an additional correction factor, leading to a modified conjecture which was eventually proved to be correct by Hooley (1967) under the assumption of the generalized Riemann hypothesis. An appropriate analogue of Artin’s primitive root conjecture may moreover be formulated for an algebraic function field K of r variables over $\mathbb{F}_{q}$. Relying on a soon to be established theorem of Weil (1948), Bilharz (1937) provided a proof in the particular case that K is a global function field (that is, r = 1), which is correct under the assumption that $g \in K$ is a geometric element. Under the same assumptions, Pappalardi and Shparlinski (1995) established a quantitative version of Bilharz’s result. In this paper we build upon these works by both generalizing to function fields in r variables over $\mathbb{F}_{q}$ and removing the assumption that $g \in K$ is geometric, thereby completing a proof of Artin’s primitive root conjecture for function fields over $\mathbb{F}_{q}$. In doing so, we moreover identify an interesting correction factor which emerges when g is not geometric. A crucial feature of our work is an exponential sum estimate over varieties that we derive from Weil’s Theorem.

L2 to Lp bounds for spectral projectors on the Euclidean two-dimensional torus

AbstractWe consider spectral projectors associated to the Euclidean Laplacian on the two-dimensional torus, in the case where the spectral window is narrow. Bounds for their L2 to Lp operator norm are derived, extending the classical result of Sogge; a new question on the convolution kernel of the projector is introduced. The methods employed include $\ell^2$ decoupling, small cap decoupling and estimates of exponential sums.

Decoupling inequalities for quadratic forms

We prove sharp $\ell^q L^p$ decoupling inequalities for $p,q \in [2,\infty)$ and arbitrary tuples of quadratic forms. Connections to prior results on decoupling inequalities for quadratic forms are also explained. We also include some applications of our results to exponential sum estimates and to Fourier restriction estimates. The proof of our main result is based on scale-dependent Brascamp--Lieb inequalities.

On a multi-parameter variant of the Bellow–Furstenberg problem

Abstract We prove convergence in norm and pointwise almost everywhere on $L^p$ , $p\in (1,\infty )$ , for certain multi-parameter polynomial ergodic averages by establishing the corresponding multi-parameter maximal and oscillation inequalities. Our result, in particular, gives an affirmative answer to a multi-parameter variant of the Bellow–Furstenberg problem. This paper is also the first systematic treatment of multi-parameter oscillation semi-norms which allows an efficient handling of multi-parameter pointwise convergence problems with arithmetic features. The methods of proof of our main result develop estimates for multi-parameter exponential sums, as well as introduce new ideas from the so-called multi-parameter circle method in the context of the geometry of backwards Newton diagrams that are dictated by the shape of the polynomials defining our ergodic averages.

On 𝐿¹² square root cancellation for exponential sums associated with nondegenerate curves in ℝ⁴

We prove sharpL12L^{12}estimates for exponential sums associated with nondegenerate curves inR4\mathbb {R}^4. We place Bourgain’s seminal result [J. Amer. Math. Soc. 30 (2017), pp. 205–224] in a larger framework that contains a continuum of estimates of different flavor. We enlarge the spectrum of methods by combining decoupling with quadratic Weyl sum estimates, to address new cases of interest. All results are proved in the general framework of real analytic curves.

Improvements of 𝑝-adic estimates of exponential sums

Let n , r n, r and f f be positive integers. Let p p be a prime number and ψ \psi be an arbitrary fixed nontrivial additive character of the finite field F q \mathbb F_q with q = p f q=p^f elements. Let F F be a polynomial in F q [ x 1 , … , x n ] \mathbb F_q[x_1,\dots ,x_n] and V V be the affine algebraic variety defined over F q \mathbb {F}_q by the simultaneous vanishing of the polynomials { F i } i = 1 r ⊆ F q [ x 1 , … , x n ] \{F_i\}_{i=1}^r\subseteq \mathbb F_q[x_1,\dots ,x_n] . Let Z ≥ 0 \mathbb {Z}_{\ge 0} stand for the set of all nonnegative integers and A A be an arbitrary nonempty subset of { 1 , … , n } \{1,\dots ,n\} . For a polynomial H ( X ) = ∑ d α d X d H(X)=\sum _{{\mathbf {d}}}\alpha _{\mathbf {d}}X^{\mathbf {d}} with d = ( d 1 , … , d n ) ∈ Z ≥ 0 n , X d = x 1 d 1 … x n d n {\mathbf {d}}=(d_1,\dots ,d_n)\in \mathbb {Z}_{\ge 0}^n, X^{\mathbf {d}}=x_1^{d_1}\dots x_n^{d_n} and α d ∈ F q ∗ \alpha _{\mathbf {d}}\in \mathbb {F}_q^* , we define deg A ⁡ ( H ) = max d { ∑ i ∈ A d i } \deg _A(H)=\max _{{\mathbf {d}}}\{\sum _{i\in A}d_i\} to be the A A -degree of H H . In this paper, for the exponential sum S ( F , V , ψ ) = ∑ X ∈ V ( F q ) ψ ( F ( X ) ) S(F,V,\psi )=\sum _{X\in V(\mathbb {F}_q)}\psi (F(X)) with V ( F q ) V(\mathbb {F}_q) being the set of the F q \mathbb {F}_q -rational points of V V , we show that o r d q S ( F , V , ψ ) ≥ | A | − ∑ i = 1 r deg A ⁡ ( F i ) max 1 ≤ i ≤ r { deg A ⁡ ( F ) , deg A ⁡ ( F i ) } \begin{equation*} \mathrm {ord}_q S(F,V,\psi )\ge \frac {|A|-\sum _{i=1}^r\deg _A(F_i)} {\max _{1\le i\le r}\{\deg _A(F),\deg _A(F_i)\}} \end{equation*} if deg A ⁡ ( F ) > 0 \deg _A(F)>0 or deg A ⁡ ( F i ) > 0 \deg _A(F_i)>0 for some i ∈ { 1 , … , r } i\in \{1,\dots ,r\} . This estimate improves Sperber’s theorem obtained in 1986. This also leads to an improvement of the p p -adic valuation of the number N ( V ) N(V) of F q \mathbb {F}_q -rational points on the variety V V which strengthens the Ax-Katz theorem. Moreover, we use the A A -degree and p p -weight A A -degree to establish p p -adic estimates on multiplicative character sums and twisted exponential sums which improve Wan’s results gotten in 1995.

On multiplicative energy of subsets of varieties

AbstractWe obtain a nontrivial upper bound for the multiplicative energy of any sufficiently large subset of a subvariety of a finite algebraic group. We also find some applications of our results to the growth of conjugates classes, estimates of exponential sums, and restriction phenomenon.

Binary sequences and lattices constructed by discrete logarithms

<abstract><p>In 1997, Mauduit and Sárközy first introduced the measures of pseudorandomness for binary sequences. Since then, many pseudorandom binary sequences have been constructed and studied. In particular, Gyarmati presented a large family of pseudorandom binary sequences using the discrete logarithms. Ten years later, to satisfy the requirement from many applications in cryptography (e.g., in encrypting "bit-maps'' and watermarking), the definition of binary sequences is extended from one dimension to several dimensions by Hubert, Mauduit and Sárközy. They introduced the measure of pseudorandomness for this kind of several-dimension binary sequence which is called binary lattices. In this paper, large families of pseudorandom binary sequences and binary lattices are constructed by both discrete logarithms and multiplicative inverse modulo $ p $. The upper estimates of their pseudorandom measures are based on estimates of either character sums or mixed exponential sums.</p></abstract>

Multilinear exponential sums with a general class of weights

In this paper we obtain some new estimates for multilinear exponential sums in prime fields with a more general class of weights than previously considered. Our techniques are based on some recent progress of Shkredov on multilinear sums which has roots in Rudnev's point plane incidence bound. We apply our estimates to obtain new results concerning exponential sums with sparse polynomials and Weyl sums over small generalized arithmetic progressions.

DIOPHANTINE INEQUALITIES OF FRACTIONAL DEGREE

This paper is concerned with the study of diagonal Diophantine inequalities of fractional degree θ, where θ > 2 is real and non-integral. For fixed non-zero real numbers λ i not all of the same sign, we write F ( x ) = λ 1 x 1 θ + ⋯ + λ s x s θ . For a fixed positive real number τ, we give an asymptotic formula for the number of positive integer solutions of the inequality | F ( x ) | < τ inside a box of side length P. Moreover, we investigate the problem of representing a large positive real number by a positive definite generalised polynomial of the above shape. A key result in our approach is an essentially optimal mean value estimate for exponential sums involving fractional powers of integers.

On the correlation measures of orders $ 3 $ and $ 4 $ of binary sequence of period $ p^2 $ derived from Fermat quotients

Let $ p $ be a prime and let $ n $ be an integer with $ (n, p) = 1 $. The Fermat quotient $ q_p(n) $ is defined as \begin{document}$ q_p(n)\equiv \frac{n^{p-1}-1}{p} \ (\bmod\ p), \quad 0\leq q_p(n)\leq p-1. $\end{document} We also define $ q_p(kp) = 0 $ for $ k\in \mathbb{Z} $. Chen, Ostafe and Winterhof constructed the binary sequence $ E_{p^2} = \left(e_0, e_1, \cdots, e_{p^2-1}\right)\in \{0, 1\}^{p^2} $ as$ \begin{equation*} \begin{split} e_{n} = \left\{\begin{array}{ll} 0, &amp; \hbox{if }\ 0\leq \frac{q_p(n)}{p}&lt;\frac{1}{2}, \\ 1, &amp; \hbox{if }\ \frac{1}{2}\leq \frac{q_p(n)}{p}&lt;1, \end{array} \right. \end{split} \end{equation*} $and studied the well-distribution measure and correlation measure of order $ 2 $ by using estimates for exponential sums of Fermat quotients. In this paper we further study the correlation measures of the sequence. Our results show that the correlation measure of order $ 3 $ is quite good, but the $ 4 $-order correlation measure of the sequence is very large.

P-adic estimates of exponential sums on curves

The purpose of this article is to prove a ``Newton over Hodge'' result for exponential sums on curves. Let $X$ be a smooth proper curve over a finite field $\mathbb{F}_q$ of characteristic $p\geq 3$ and let $V \subset X$ be an affine curve. For a regular function $\overline{f}$ on $V$, we may form the $L$-function $L(\overline{f},V,s)$ associated to the exponential sums of $\overline{f}$. In this article, we prove a lower estimate on the Newton polygon of $L(\overline{f},V,s)$. The estimate depends on the local monodromy of $f$ around each point $x \in X-V$. This confirms a hope of Deligne that the irregular Hodge filtration forces bounds on $p$-adic valuations of Frobenius eigenvalues. As a corollary, we obtain a lower estimate on the Newton polygon of a curve with an action of $\mathbb{Z}/p\mathbb{Z}$ in terms of local monodromy invariants.

On the dynamical system generated by the Möbius transformation at prime times

We study the distribution of the sequence of elements of the discrete dynamical system generated by iterations of the Möbius map \(x \mapsto (ax + b)/(cx+d)\) over a finite field of p elements at the moments of time that correspond to prime numbers. In particular, we obtain nontrivial estimates of exponential sums with such sequences.

Estimates of short exponential sums with primes in major arcs

Estimates of short exponential sums with primes in major arcs

The remark on the mean value theorem for the absolute value of Dirichlet L-function in the critical strip

We continue our researches concerning the generalization and improvement of R.T.Turganaliev’s result that states an asymptotic formula for the mean value of the Riemann zeta function in the critical strip with power factor saving in the remainder term.We find an asymptotic for the mean value of Dirichlet L-function in the critical strip. This assertion improves R.T.Turganaliev’s theorem for zeta-function in the whole interval $$(1/2 < Re 𝑠 ≤ 1)$$. Our result is based on the special use of the estimation of exponential sums by second derivative test.

Small cap decouplings

We develop a toolbox for proving decouplings into boxes with diameter smaller than the canonical scale. As an application of this new technique, we solve three problems for which earlier methods have failed. We start by verifying the small cap decoupling for the parabola. Then we find sharp estimates for exponential sums with small frequency separation on the moment curve in $$\mathbb {R}^3$$ . This part of the work relies on recent improved Kakeya-type estimates for planar tubes, as well as on new multilinear incidence bounds for plates and planks. We also combine our method with the recent advance on the reverse square function estimate, in order to prove small cap decoupling into square-like caps for the two dimensional cone. The Appendix by Roger Heath-Brown contains an application of the new exponential sum estimates for the moment curve, to the Riemann zeta-function.

Hua’s theorem on five squares of primes

We give an alternative proof of Hua’s theorem that each large N ≡ 5 (mod 24) can be written as a sum of five squares of primes. The proof depends on an estimate of exponential sums involving the Mobius function.

Mean Values of Arithmetic Functions under Congruences with the Euler Function

We examine the average order of some arithmetic functions written as sums over Euler function in arithmetic progression and in general over such that is a prime number, an integer and is a polynomial function with integer coefficients and a degree that is not constant modulo Our results are based on various estimates of rational exponential sums with the Euler Function in arithmetic progression which are due to William Banks and Igor E. Shparlinski.

Positive-Definite Functions, Exponential Sums and the Greedy Algorithm: a Curious Phenomenon

We describe a curious dynamical system that results in sequences of real numbers in [0,1] with seemingly remarkable properties. Let the even function f:T→R satisfy f̂(k)≥c|k|−2 and define a sequence via xn=argminx∑k=1n−1f(x−xk). Such sequences (xn)n=1∞ seem to be astonishingly regularly distributed in various ways (satisfying favorable exponential sum estimates; every interval J⊂[0,1] contains ∼|J|n elements). We prove W2μ,ν≤cn,whereμ=1n∑k=1nδxk is the empirical distribution, ν=dx is the Lebesgue measure and W2(μ,ν) is the 2-Wasserstein distance between these two. Much stronger results seem to be true and it is an interesting problem to understand this dynamical system better. We obtain optimal results in dimension d≥3: using G(x,y) to denote the Green’s function of the Laplacian on a compact manifold, we show that xn=argminx∈M∑k=1n−1G(x,xk)satisfiesW21n∑k=1nδxk,dx≲1n1∕d.

Littlewood’s Problem for Sets with Multidimensional Structure

Abstract We give $L^1$-norm estimates for exponential sums of a finite sets $A$ consisting of integers or lattice points. Under the assumption that $A$ possesses sufficient multidimensional structure, our estimates are stronger than those of McGehee–Pigno–Smith and Konyagin. These theorems improve upon past work of Petridis.