Estimates For Sums Research Articles

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.

Об одной теореме о среднем значении кратных тригонометрических сумм

Доказана теорема о среднем для кратных тригонометрических сумм, обобщающая теорему Г. И. Архипова [12, 13]. Первая теорема подобного типа лежит в сердцевине метода И. М. Виноградова [2]. В работе найден вариант теоремы с "равноправными" длинами промежутков изменения переменных. Интересным приложением метода И. М. Виноградова являются оценки дзетовых сумм вида$$\sum_{n\leq P}n^{it}.$$Подобным приложением теоремы о среднем, доказанной нами, служат оценки сумм вида$$\sum_{n\leq P_1}\dots\sum_{n\leq P_r}(n_1\dots n_r+k)^{it}, \sum_{n\leq P}\tau_s(n)(n+k)^{it}, \sum_{p\leq P}(p+k)^{it}.$$

New estimates for exponential sums over multiplicative subgroups and intervals in prime fields

Let H be a multiplicative subgroup of Fp⁎ of order H>p1/4. We show thatmax(a,p)=1⁡|∑x∈Hep(ax)|≤H1−31/2880+o(1), where ep(z)=exp⁡(2πiz/p), which improves a result of Bourgain and Garaev (2009). We also obtain new estimates for double exponential sums with product nx with x∈H and n∈N for a short interval N of consecutive integers.

The Lindelöf hypothesis for primes is equivalent to the Riemann hypothesis

We recast the classical Lindelöf hypothesis as an estimate for the sums ∑ n ≤ x n − i t \sum _{n\leq x}n^{-it} . This leads us to propose that a more general form of the Lindelöf hypothesis may be true, one involving estimates for sums of the type ∑ n ≤ x n ∈ N n − i t , \begin{equation*} \sum _{ \substack {n\leq x \\ n\in \mathscr {N} }}n^{-it}, \end{equation*} where N \mathscr {N} can be a quite general sequence of real numbers. We support this with several examples and show that when N = P \mathscr {N}=\mathbb {P} , the sequence of prime numbers, the truth of our conjecture is equivalent to the Riemann hypothesis. Moreover, if our conjecture holds for N = P ( a , q ) \mathscr {N}=\mathbb {P}(a, q) , the primes congruent to a ( mod q ) a \pmod q , with a a coprime to q q , then the Riemann hypothesis holds for all Dirichlet L L -functions with characters modulo q q , and conversely. These results suggest that a general form of the Lindelöf hypothesis may be both true and more fundamental than the classical Lindelöf hypothesis and the Riemann hypothesis.

Теорема о среднем значении тригонометрических сумм на последовательности многочленов биномиального типа

Доказана теорема о среднем для тригонометрических сумм на последовательности многочленов биномиального типа. Как известно, классическая теорема И. М. Виноградова о среднем [10] относится к последовательности многочленов вида $\{x^n, n\geq 0\}.$Важным приложением найденной теоремы о среднем являются оценки сумм вида$$\sum_{m\leq P}e^{2\pi if(m)}, f(m)=\sum_{k=0}^n\alpha_kp_k(m),$$где $p_k(x)$ - последовательность целозначных многочленов биномиального типа,а набор чисел $(\alpha_1\alpha_1,\dots,\alpha_n)$ представляет собой точку $n$-мерного единичного куба $\Omega: 0\leq \alpha_1,\dots,$ $\alpha_n<1.$

Exponential sums of squares of Fourier coefficients of cusp forms

We prove nontrivial estimates for linear sums of squares of Fourier coefficients of holomorphic and Maass cusp forms twisted by additive characters. For holomorphic forms f, we show that if $$|\alpha -a/q| \le 1/q^2$$ with $$(a,q)=1$$ , then for any $$\varepsilon >0$$ , $$\begin{aligned} \qquad \qquad \sum _{n\leqslant X}{\lambda _f(n)}^2 e(n\alpha ) \ll _{f, \varepsilon } X^{{\frac{4}{5}}+\varepsilon } \quad \text {for} \ X^{{\frac{1}{5}}} \ll q \ll X^{{\frac{4}{5}}}. \end{aligned}$$ Moreover, for any $$\varepsilon > 0,$$ there exists a set $$S \subset (0, 1)$$ with $$\mu (S)=1$$ such that for every $$\alpha \in S$$ , there exists $$X_0=X_0(\alpha )$$ such that the above inequality holds true for any $$\alpha \in S$$ and $$X \geqslant X_0(\alpha ).$$ A weaker bound for Maass cusp forms is also established.

The Quadratic Residues and Some of Their New Distribution Properties

In this paper, we give some interesting identities and asymptotic formulas for one kind of counting function, by studying the computational problems involving the symmetry sums of one kind quadratic residues and quadratic non-residues mod p . The main methods we used are the properties of the Legendre’s symbol mod p , and the estimate for character sums. As application, we solve two open problems proposed by Zhiwei Sun.

Is building energy simulation based on TMY representative: A comparative simulation study on doe reference buildings in Toronto with typical year and historical year type weather files

The goal of this study is to determine whether the use of Typical Year (TY) type weather file is adequate for estimating heating and cooling load in Building Energy Simulation (BES). Two Typical Year (TY) Type weather files – Canadian Weather Year for Energy Calculation(CWEC) 1990s and CWEC 2016 are compared with Historical Year (HY) weather data Canadian Weather Energy and Engineering Datasets(CWEEDs) 1998 to 2014. Three stages of analysis are completed – 1) comparison of the cumulative distribution of Dry Bulb Temperature and Global Horizontal Irradiance; 2) analysis of extreme weather event occurrences through comparing temperature-based weather indices; 3) comparison of BES results from simulating DOE prototype buildings. Results show a hotter summer and warmer winter in CWEC 2016 indicated by a shift of median temperature in stage 1, increased hot weather characterizing indices and decreased cold weather characterizing indices in stage 2 when compared with CWEC 1990s. This translates to a lower heating energy use and higher cooling energy use in stage 3 with simulation using CWEC 2016. The total 17-year heating and cooling energy sum estimation using CWEC 2016 shows lower percentage difference for all buildings, ranging from 0 to -8% difference, where most buildings are -1% or -2% from the estimation using historical weather data. However, neither simulation using CWEC 1990s nor CWEC 2016 yield consistent accuracy when compared to the yearly Historical simulation results, since the occurrence of years with hotter/colder temperature shows a periodical cycle, some years have more hot weather events, whereas other years have more cold weather event. This study concludes that Typical Year simulation alone is not sufficient for Building Energy Simulation because 1) past weather data might not be a good indication of recent or future weather events due to changes in climate; 2) the update cycle of TY type weather files are infrequent and therefore most recent climate may not be included in the analysis. Future work should focus on the development of a commonly accepted practice in using multi-year simulation, as well as a commonly accepted format for future or stochastic weather file format.

Congruences with intervals and arbitrary sets

Given a prime p, an integer $$H\in [1,p)$$, and an arbitrary set $${\mathcal {M}} \subseteq {\mathbb {F}} _p^*$$, where $${\mathbb {F}} _p$$ is the finite field with p elements, let $$J(H,{\mathcal {M}} )$$ denote the number of solutions to the congruence $$\begin{aligned} xm\equiv yn~\mathrm{mod}~ p \end{aligned}$$for which $$x,y\in [1,H]$$ and $$m,n\in {\mathcal {M}} $$. In this paper, we bound $$J(H,{\mathcal {M}} )$$ in terms of p, H, and the cardinality of $${\mathcal {M}} $$. In a wide range of parameters, this bound is optimal. We give two applications of this bound: to new estimates of trilinear character sums and to bilinear sums with Kloosterman sums, complementing some recent results of Kowalski et al. (Stratification and averaging for exponential sums: bilinear forms with generalized Kloosterman sums, 2018, arXiv:1802.09849).

Decoupling for moment manifolds associated to Arkhipov–Chubarikov–Karatsuba systems

We prove ℓpLp decoupling inequalities for a class of moment manifolds. These inequalities imply optimal mean value estimates for multidimensional Weyl sums of the kind considered by Arkhipov, Chubarikov, and Karatsuba and by Parsell.In our proofs we take a new point of view on the Bourgain–Demeter–Guth induction on scales argument. This point of view substantially simplifies even the proof of ℓ2Lp decoupling for the moment curve.

Uniform bounds of the Piltz divisor problem over number fields

We consider the upper bound of Piltz divisor problem over number fields. Piltz divisor problem is known as a generalization of the Dirichlet divisor problem. We deal with this problem over number fields and improve the error term of this function for many cases. Our proof uses the estimate of exponential sums. We also show uniform results for ideal counting function and relatively $r$-prime lattice points as one of applications.

Poissonian pair correlation in higher dimensions

Let (xn)n=1∞ be a sequence on the torus T (normalized to length 1). A sequence (xn) is said to have Poissonian pair correlation if, for all s>0,limN→∞⁡1N#{1≤m≠n≤N:|xm−xn|≤sN}=2s. It is known that this implies uniform distribution of the sequence (xn). Hinrichs, Kaltenböck, Larcher, Stockinger & Ullrich extended this result to higher dimensions and showed that sequences (xn) in [0,1]d that satisfy, for all s>0,limN→∞⁡1N#{1≤m≠n≤N:‖xm−xn‖∞≤sN1/d}=(2s)d are also uniformly distributed. We prove the same result for the extension by the Euclidean norm: if a sequence (xn) in Td satisfies, for all s>0,limN→∞⁡1N#{1≤m≠n≤N:‖xm−xn‖2≤sN1/d}=ωdsd, where ωd is the volume of the unit ball, then (xn) is uniformly distributed. Our approach shows that Poissonian Pair Correlation implies an exponential sum estimate that resembles and implies the Weyl criterion.

Further improvement on bounds for L-functions related to GL(3)

Let [Formula: see text] be a fixed self-dual Hecke–Maass form for [Formula: see text], and let [Formula: see text] be an even Hecke–Maass form for [Formula: see text] with Laplace eigenvalue [Formula: see text], [Formula: see text]. A subconvexity bound for [Formula: see text] is improved to [Formula: see text], and a subconvexity bound for [Formula: see text] is improved to [Formula: see text]. New techniques employed include an application of an asymptotic formula by Salazar and Ye [Spectral square moments of a resonance sum for Maass forms, Front. Math. China 12(5) (2017) 1183–1200] to make error terms negligible, an iterative algorithm to locate stationary point, and a non-trivial estimation of Kloosterman sums.

Estimates for sums of eigenvalues of the free plate via the fourier transform

<p style='text-indent:20px;'>We obtain estimates for sums of eigenvalues of the free plate under tension in terms of the dimension of the ambient space, the volume of the domain, and the tension parameter. We consequently obtain similar estimates for the eigenvalues. Our results generalize those of Kröger for the free membrane contained in [<xref ref-type="bibr" rid="b16">16</xref>].

On Sums of Products in $$\mathbb{F}_p \times \mathbb{F}_p$$

The problem of sums of products in $$\mathbb{F}_p \times \mathbb{F}_p$$ is considered. An estimate for sums of products improving Bourgain’s result of 2005 is obtained. This estimate is applied to the problem of estimating polynomial exponential sums over multiplicative subgroups in $$\mathbb{F}_p^*$$ .

Means of some arithmetical functions on shifted smooth numbers in arithmetic progression

We provide asymptotic estimates for sums of some arithmetical functions taken over shifted smooth numbers in arithmetic progression related to the sum of digits function. Our results are based on various estimates on the distribution of smooth numbers in arithmetic progressions which are due to G Tenenbaum, Stephanie S Loiperdinger and Igor E Shparlinski.

A logarithmic estimate for harmonic sums and the digamma function, with an application to the Dirichlet divisor problem

Let H_{n} = sum_{r=1}^{n} 1/r and H_{n}(x) = sum_{r=1}^{n} 1/(r+x). Let psi(x) denote the digamma function. It is shown that H_{n}(x) + psi(x+1) is approximated by frac{1}{2}log f(n+x), where f(x) = x^{2} + x + frac{1}{3}, with error term of order (n+x)^{-5}. The cases x = 0 and n = 0 equate to estimates for H_{n} - gamma and psi(x+1) itself. The result is applied to determine exact bounds for a remainder term occurring in the Dirichlet divisor problem.

An exponential estimate for the cubic partial sums of multiple Fourier series

We prove an exponential integral estimate for the cubic partial sums of multiple Fourier series on sets of large measure. This estimate yields some new properties of Fourier series.

Menshov Type Correction Theorems for Sequences of Compact Operators

We prove Menshov type "correction" theorems for sequences of compact operators, recovering several results of Fourier series in trigonometric and Walsh systems. The paper clarifies the main ingredient, which is important in the study of such "correction" theorems. That is the weak-$L^1$ estimate for the maximal Fourier sums of indicator functions of some specific sets.