Vinogradov's Mean Value Theorem Research Articles

A quadratic Vinogradov mean value theorem in finite fields

A quadratic Vinogradov mean value theorem in finite fields

Weyl Sums over Integers with Digital Restrictions

We estimate Weyl sums over the integers with sum of binary digits either fixed or restricted by some congruence condition. In our proofs, we use ideas that go back to a paper by Banks, Conflitti, and the first author (2002). Moreover, we apply the “main conjecture” on the Vinogradov mean value theorem established by Bourgain, Demeter, and Guth (2016) and Wooley (2016, 2019). We use our result to give an estimate of the discrepancy of point sets defined by the values of polynomials at arguments having the sum of binary digits restricted in different ways.

A decoupling interpretation of an old argument for Vinogradov's Mean Value Theorem

AbstractWe interpret into decoupling language a refinement of a 1973 argument due to Karatsuba on Vinogradov's mean value theorem. The main goal of our argument is to answer what precisely solution counting in older partial progress on Vinogradov's mean value theorem corresponds to in Fourier decoupling theory.

Distribution of recursive matrix pseudorandom number generator modulo prime powers

Given a matrix A ∈ G L d ( Z ) A\in \mathrm {GL}_d(\mathbb {Z}) . We study the pseudorandomness of vectors u n \mathbf {u}_n generated by a linear recurrence relation of the form u n + 1 ≡ A u n ( mod p t ) , n = 0 , 1 , … , \begin{equation*} \mathbf {u}_{n+1} \equiv A \mathbf {u}_n \pmod {p^t}, \qquad n = 0, 1, \ldots , \end{equation*} modulo p t p^t with a fixed prime p p and sufficiently large integer t ⩾ 1 t \geqslant 1 . We study such sequences over very short segments of length which has not been accessible via previously used methods. Our technique is based on the method of N. M. Korobov [Mat. Sb. (N.S.) 89(131) (1972), pp. 654–670, 672] of estimating double Weyl sums and a fully explicit form of the Vinogradov mean value theorem due to K. Ford [Proc. London Math. Soc. (3) 85 (2002), pp. 565–633]. This is combined with some ideas from the work of I. E. Shparlinski [Proc. Voronezh State Pedagogical Inst., 197 (1978), 74–85 (in Russian)] which allows us to construct polynomial representations of the coordinates of u n \mathbf {u}_n and control the p p -adic orders of their coefficients in polynomial representations.

On a hybrid version of the Vinogradov mean value theorem

Given a family $$\varphi = (\varphi_1, \ldots, \varphi_d)\in \mathbb{Z}[T]^d$$ of d distinct nonconstant polynomials, a positive integer $$k\le d$$ and a real positive parameter $$\rho$$ , we consider the mean value $$M_{k, \rho} (\varphi, N) = \int_{{\rm x} \in [0,1]^k} \sup_{{\rm y} \in [0,1]^{d-k}} | S_{\varphi}({\rm x}, {\rm y}; N) |^\rho \,d{\rm x} $$ of exponential sums $$S_{\varphi}({\rm x}, {\rm y}; N) = \sum_{n=1}^{N} \exp\biggl(2 \pi i\biggl(\sum_{j=1}^k x_j \varphi_j(n)+ \sum_{j=1}^{d-k}y_j\varphi_{k+j}(n)\biggr)\biggr), $$ where $${\rm x} = (x_1, \ldots, x_k)$$ and $${\rm y} =(y_1, \ldots, y_{d-k})$$ . The case of polynomials $$\varphi_i(T) = T^i, i =1, \ldots, d$$ and $$k=d$$ corresponds to the classical Vinaogradov mean value theorem. Here motivated by recent works of Wooley [14] and the authors [9] on bounds on $${\rm sup}_{{\rm y} \in [0,1]^{d-k}} | S_{\varphi}({\rm x}, {\rm y}; N) |$$ for almost all $${\rm x} \in [0,1]^k$$ , we obtain nontrivial bounds on $$M_{k, \rho} (\varphi, N)$$ .

Points on polynomial curves in small boxes modulo an integer

Given an integer q and a polynomial f∈Zq[X] of degree d with coefficients in the residue ring Zq=Z/qZ, we obtain new results concerning the number of solutions to congruences of the form y≡f(x)(modq), with integer variables lying in some cube B of side length H. Our argument uses ideas of Cilleruelo, Garaev, Ostafe and Shparlinski which reduces the problem to the Vinogradov mean value theorem and a lattice point counting problem. We treat the lattice point problem differently, using transference principles from the geometry of numbers. We also use a variant of the main conjecture for the Vinogradov mean value theorem of Bourgain, Demeter and Guth and of Wooley, which allows one to deal with solutions to the Vinogradov mean value theorem when the variables run through rather sparse sets.

Restricted Mean Value Theorems and the Metric Theory of Restricted Weyl Sums

Abstract We study the behaviour of Weyl sums on a subset ${\mathcal X}\subseteq [0,1)^d$ with a natural measure µ on ${\mathcal X}$. For certain measure spaces $({\mathcal X}, \mu),$ we obtain non-trivial bounds for the mean values of the Weyl sums, and for µ-almost all points of ${\mathcal X}$ the Weyl sums satisfy the square root cancellation law. Moreover, we characterize the size of the exceptional sets in terms of Hausdorff dimension. Finally, we derive variants of the Vinogradov mean value theorem averaging over measure spaces $({\mathcal X}, \mu)$. We obtain general results, which we refine for some special spaces ${\mathcal X}$ such as spheres, moment curves and line segments.

Improving Estimates for Discrete Polynomial Averages

For a polynomial $P$ mapping the integers into the integers, define an averaging operator $A_{N} f(x):=\frac{1}{N}\sum_{k=1}^N f(x+P(k))$ acting on functions on the integers. We prove sufficient conditions for the $\ell^{p}$-improving inequality \begin{equation*} \|A_N f\|_{\ell^q(\mathbb{Z})} \lesssim_{P,p,q} N^{-d(\frac{1}{p}-\frac{1}{q})} \|f\|_{\ell^p(\mathbb{Z})}, \qquad N \in\mathbb{N}, \end{equation*} where $1\leq p \leq q \leq \infty$. For a range of quadratic polynomials, the inequalities established are sharp, up to the boundary of the allowed pairs of $(p,q)$. For degree three and higher, the inequalities are close to being sharp. In the quadratic case, we appeal to discrete fractional integrals as studied by Stein and Wainger. In the higher degree case, we appeal to the Vinogradov Mean Value Theorem, recently established by Bourgain, Demeter, and Guth.

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 $\Omega: 0\leq \alpha_1,\dots,$ $\alpha_n<1.$

Bounds for discrete moments of Weyl sums and applications

We prove two bounds for discrete moments of Weyl sums. The first one can be obtained using a standard approach. The second one involves an observation how this method can be improved, which leads to a sharper bound in certain ranges. The proofs both build on the recently proved main conjecture for Vinogradov's mean value theorem. We present two selected applications: First, we prove a new $k$- th derivative test for the number of integer points close to a curve by an exponential sum approach. This yields a stronger bound than existing results obtained via geometric methods, but it is only applicable for specific functions. As second application we prove a new improvement of the polynomial large sieve inequality for one-variable polynomials of degree $k \geq 4$.

Distribution of short subsequences of inversive congruential pseudorandom numbers modulo $2^t$

In this paper we study the distribution of very short sequences of inversive congruential pseudorandom numbers modulo $2^t$. We derive a new bound on exponential sums with such sequences and use it to give estimate their discrepancy. The technique we use, based the method of N. M. Korobov (1972) of estimating double Weyl sums and a fully explicit form of the Vinogradov mean value theorem due to K. Ford (2002), has never been used in this area and is very likely to find further applications.

Effective Vinogradov's mean value theorem via efficient boxing

We combine Wooley's efficient congruencing method with earlier work of Vinogradov and Hua to get effective bounds on Vinogradov's mean value theorem.

The Vinogradov Mean Value Theorem (after Wooley, and Bourgain, Demeter and Guth)

This is the expository essay that accompanies my Bourbaki Seminar on 17 June 2017 on the landmark proof of the Vinogradov Mean Value Theorem, and the two approaches developed in the work of Wooley and of Bourgain, Demeter and Guth.

Nested efficient congruencing and relatives of Vinogradov's mean value theorem

We apply a nested variant of multigrade efficient congruencing to estimate mean values related to that of Vinogradov. We show that when $\varphi_j\in \mathbb Z[t]$ $(1\le j\le k)$ is a system of polynomials with non-vanishing Wronskian, and $s\le k(k+1)/2$, then for all complex sequences $(\mathfrak a_n)$, and for each $\epsilon>0$, one has \[ \int_{[0,1)^k} \left| \sum_{|n|\le X} {\mathfrak a}_n e(\alpha_1\varphi_1(n)+\ldots +\alpha_k\varphi_k(n)) \right|^{2s} {\rm d}{\boldsymbol \alpha} \ll X^\epsilon \left( \sum_{|n|\le X} |{\mathfrak a}_n|^2\right)^s. \] As a special case of this result, we confirm the main conjecture in Vinogradov's mean value theorem for all exponents $k$, recovering the recent conclusions of the author (for $k=3$) and Bourgain, Demeter and Guth (for $k\ge 4$). In contrast with the $l^2$-decoupling method of the latter authors, we make no use of multilinear Kakeya estimates, and thus our methods are of sufficient flexibility to be applicable in algebraic number fields, and in function fields. We outline such extensions.

Incomplete gauss sums modulo primes

We obtain a new bound for incomplete Gauss sums modulo primes. Our argument falls under the framework of Vinogradov's method which we use to reduce the problem under consideration to bounding the number of solutions to two distinct systems of congruences. The first is related to Vinogradov's mean value theorem, although the second does not appear to have been considered before. Our bound improves on current results in the range $N\ge q^{2k^{-1/2}+O(k^{-3/2})}$.

On Bounds for Characteristic Functions of Powers of Asymptotically Normal Variables

We obtain upper bounds for the absolute values of the characteristic functions of the $k$th powers of asymptotically normal random variables. Estimates are proved for the case when asymptotically normal random variables are normalized sums of independent identically distributed summands with a “regular” distribution. Possible generalizations are considered. The estimates extend the results of previous studies, where for the distributions of the summands, the presence of either a discrete or an absolutely continuous component was required. The proofs of the bounds are based on the stochastic generalization of the I. M. Vinogradov mean value theorem, which is also obtained in the present paper.

A Bombieri--Vinogradov Theorem with products of Gaussian primes as moduli

We prove a version of the Bombieri--Vinogradov Theorem with certain products of Gaussian primes as moduli, making use of their special form as polynomial expressions in several variables. Adapting Vaughan's proof of the classical Bombieri--Vinogadov Theorem, cp. [10] to this setting, we apply the polynomial large sieve inequality that has been proved in [7] and which includes recent progress in Vinogradov's mean value theorem due to Parsell \emph{et al.} in [9]. From the benefit of these improvements, we obtain an extended range for the variables compared to the range obtained from standard arguments only.

APPROXIMATING THE MAIN CONJECTURE IN VINOGRADOV'S MEAN VALUE THEOREM

We apply multigrade efficient congruencing to estimate Vinogradov's integral of degree $k$ for moments of order $2s$, establishing strongly diagonal behaviour for $1\le s\le \frac{1}{2}k(k+1)-\frac{1}{3}k+o(k)$. In particular, as $k\rightarrow \infty$, we confirm the main conjecture in Vinogradov's mean value theorem for 100% of the critical interval $1\le s\le \frac{1}{2}k(k+1)$.

Proof of the main conjecture in Vinogradov's Mean Value Theorem \break for degrees higher than three

We prove the main conjecture in Vinogradov's Mean Value Theorem for degrees higher than three. This will be a consequence of a sharp decoupling inequality for curves

On the Waring–Goldbach problem for eighth and higher powers

Recent progress on Vinogradov's mean value theorem has resulted in improved estimates for exponential sums of Weyl type. We apply these new estimates to obtain sharper bounds for the function $H(k)$ in the Waring--Goldbach problem. We obtain new results for all exponents $k\ge 8$, and in particular establish that $H(k)\le (4k-2)\log k+k-7$ when $k$ is large, giving the first improvement on the classical result of Hua from the 1940s.