Efficient Congruencing Research Articles

Diameter‐free estimates for the quadratic Vinogradov mean value theorem

Let s ⩾ 3 $s \geqslant 3$ be a natural number, let ψ ( x ) $\psi (x)$ be a polynomial with real coefficients and degree d ⩾ 2 $d \geqslant 2$ , and let A $A$ be some large, non-empty, finite subset of real numbers. We use E s , 2 ( A ) $E_{s,2}(A)$ to denote the number of solutions to the system of equations ∑ i = 1 s ( ψ ( x i ) − ψ ( x i + s ) ) = ∑ i = 1 s ( x i − x i + s ) = 0 , \begin{equation*}\hskip2.5pc \sum _{i=1}^{s} (\psi (x_i) - \psi (x_{i+s}) )= \sum _{i=1}^{s} (x_i - x_{i+s} ) = 0, \end{equation*} where x i ∈ A $x_i \in A$ for each 1 ⩽ i ⩽ 2 s $1 \leqslant i \leqslant 2s$ . Our main result shows that E s , 2 ( A ) ≪ d , s | A | 2 s − 3 + η s , \begin{equation*}\hskip6pc E_{s,2}(A) \ll _{d,s} |A|^{2s -3 + \eta _{s}}, \end{equation*} where η 3 = 1 / 2 $\eta _3 = 1/2$ , and η s = ( 1 / 4 − 1 / 7246 ) · 2 − s + 4 $\eta _{s} = (1/4- 1/7246)\cdot 2^{-s + 4}$ when s ⩾ 4 $s \geqslant 4$ . The only other previously known result of this flavour is due to Bourgain and Demeter, who showed that when ψ ( x ) = x 2 $\psi (x) = x^2$ and s = 3 $s=3$ , we have E 3 , 2 ( A ) ≪ ε | A | 3 + 1 / 2 + ε , \begin{equation*}\hskip6pc E_{3,2}(A) \ll _{\epsilon } |A|^{3 + 1/2 + \epsilon }, \end{equation*} for each ε > 0 $\epsilon > 0$ . Thus, our main result improves upon the above estimate, while also generalising it for larger values of s $s$ and more wide-ranging choices of ψ ( x ) $\psi (x)$ . The novelty of our estimates is that they only depend on d $d$ , s $s$ and | A | $|A|$ , and are independent of the diameter of A $A$ . Thus, when A $A$ is a sparse set, our results are stronger than the corresponding bounds that are provided by methods such as decoupling and efficient congruencing. Consequently, our strategy differs from these two lines of approach, and we employ techniques from incidence geometry, arithmetic combinatorics and analytic number theory. Amongst other applications, our estimates lead to stronger discrete restriction estimates for sparse sequences.

Efficient congruencing in ellipsephic sets: the general case

In this paper, we bound the number of solutions to a general Vinogradov system of equations [Formula: see text] as well as other related systems, in which the variables are required to satisfy digital restrictions in a given base. Specifically, our sets of permitted digits have the property that there are few representations of a natural number as sums of elements of the digit set — the set of squares serving as a key example. We obtain better bounds using this additive structure than could be deduced purely from the size of the set of variables. In particular, when the digits are required to be squares, we obtain diagonal behavior with [Formula: see text] variables.

Decoupling for fractal subsets of the parabola

We consider decoupling for a fractal subset of the parabola. We reduce studying \(l^{2}L^{p}\) decoupling for a fractal subset on the parabola \(\{(t, t^2) : 0 \le t \le 1\}\) to studying \(l^{2}L^{p/3}\) decoupling for the projection of this subset to the interval [0, 1]. This generalizes the decoupling theorem of Bourgain-Demeter in the case of the parabola. Due to the sparsity and fractal like structure, this allows us to improve upon Bourgain–Demeter’s decoupling theorem for the parabola. In the case when p/3 is an even integer we derive theoretical and computational tools to explicitly compute the associated decoupling constant for this projection to [0, 1]. Our ideas are inspired by the recent work on ellipsephic sets by Biggs (arXiv:1912.04351, 2019 and Acta Arith. 200(4):331–348, 2021) using nested efficient congruencing.

A bilinear proof of decoupling for the cubic moment curve

Using a bilinear method that is inspired by the method of efficient congruencing of Wooley [Woo16], we prove a sharp decoupling inequality for the moment curve in $\mathbb{R}^3$.

An $l^2$ decoupling interpretation of efficient congruencing: the parabola

We give a new proof of l^2 decoupling for the parabola inspired from efficient congruencing. Making quantitative this proof matches a bound obtained by Bourgain for the discrete restriction problem for the parabola. We illustrate similarities and differences between this new proof and efficient congruencing and the proof of decoupling by Bourgain and Demeter. We also show where tools from decoupling such as l^2L^2 decoupling, Bernstein’s inequality, and ball inflation come into play.

Efficient congruencing in ellipsephic sets: the quadratic case

In this paper, we bound the number of solutions to a quadratic Vinogradov system of equations in which the variables are required to satisfy digital restrictions in a given base. Certain sets of permitted digits, namely those giving rise to few representations of natural numbers as sums of elements of the digit set, allow us to obtain better bounds than would be possible using the size of the set alone. In particular, when the digits are required to be squares, we obtain diagonal behaviour with 12 variables.

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.

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 $1\le s\le \frac{1}{2}k(k+1)$.

The cubic case of the main conjecture in Vinogradov's mean value theorem

We apply a variant of the multigrade efficient congruencing method to estimate Vinogradov's integral of degree 3 for moments of order 2s, establishing strongly diagonal behaviour for 1⩽s⩽6. Consequently, the main conjecture is now known to hold for the first time in a case of degree exceeding 2.

Multigrade efficient congruencing and Vinogradov's mean value theorem

We develop a multigrade enhancement of the efficient congruencing method to estimate Vinogradov's integral of degree $k$ for moments of order $2s$, thereby obtaining near-optimal estimates for $\tfrac{5}{8}k^2<s\le k^2-k+1$. There are numerous applications. In particular, when $k$ is large, the anticipated asymptotic formula in Waring's problem is established for sums of $s$ $k$th powers of natural numbers whenever $s>1.543k^2$. The asymptotic formula is also established for sums of $28$ fifth powers.

Multidimensional Vinogradov-type Estimates in Function Fields

AbstractLet 𝔽q[t] denote the polynomial ring over the finite field 𝔽q. We employ Wooley's new efficient congruencing method to prove certain multidimensional Vinogradov-type estimates in 𝔽q[t]. These results allow us to apply a variant of the circle method to obtain asymptotic formulas for a system connected to the problem about linear spaces lying on hypersurfaces defined over 𝔽q[t].

Vinogradov's mean value theorem via efficient congruencing

We obtain estimates for Vinogradov's integral which for the first time approach those conjectured to be the best possible. Several applications of these new bounds are provided. In particular, the conjectured asymptotic formula in Waring's problem holds for sums of s kth powers of natural numbers whenever s is at least 2k^2+2k-3.

A NEW BOUND FOR THE LARGE SIEVE INEQUALITY WITH POWER MODULI

We give a new bound for the large sieve inequality with power moduli qk that is uniform in k. The proof uses a new theorem due to Wooley from his work [Vinogradov's mean value theorem via efficient congruencing, to appear in Ann. of Math.] on efficient congruencing.