Arithmetic combinatorics on Vinogradov systems

Mudgal, Akshat

doi:10.1090/tran/8121

Cited by 3 publications

(4 citation statements)

References 15 publications

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“…In particular, Bourgain and Demeter [2, Question 2.13] conjectured that for all finite, non‐empty sets

A

of real numbers and for all

\epsilon &gt; 0

, we have

\begin{equation} J_{3,2}(A) \ll _{\epsilon } |A|^{3 + \epsilon }. \end{equation}

Combining this with (), we see that () is equivalent to the following conjecture (see also [7, Conjecture 1.5]). Conjecture Let

s \geqslant 3

be a natural number, let

A

be a finite, non‐empty set of real numbers and let

\epsilon &gt; 0

be a real number.…”

Section: Introductionmentioning

confidence: 63%

“…These particular types of sumsets were studied in [7], and from the point of view of additive combinatorics, estimates on cardinalities of such sets are closely related to bounds for

E_{s,2}(A)

. In this paper, we record some further threshold‐breaking lower bounds for cardinalities of sumsets of the above form.…”

Section: Introductionmentioning

confidence: 99%

“…With these definitions in hand, we note that for any fixed

m \geqslant 1

, if a set

A

is well‐spaced and

m

‐dense, then () implies that

\begin{equation*} J_{s,2}(A)\ll _{s, \epsilon } X_A^{\epsilon } |A|^{2s - 3} \ll _{s, \epsilon } |A|^{2s-3+ m\epsilon }, \end{equation*}

whence, we obtain the desired conclusion by rescaling

\epsilon

appropriately. We refer the reader to [7] for a more detailed introduction to the discussion surrounding Conjecture 1.4 and estimates of the above form.…”

Section: Introductionmentioning

confidence: 99%

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Diameter‐free estimates for the quadratic Vinogradov mean value theorem

Mudgal

2022

Proceedings of London Math Soc

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Let s⩾3$s \geqslant 3$ be a natural number, let ψfalse(xfalse)$\psi (x)$ 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 Es,2(A)$E_{s,2}(A)$ to denote the number of solutions to the system of equations 30.0pt∑i=1s(ψfalse(xifalse)−ψfalse(xi+sfalse))badbreak=∑i=1s(xi−xi+s)goodbreak=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 xi∈A$x_i \in A$ for each 1⩽i⩽2s$1 \leqslant i \leqslant 2s$. Our main result shows that 72.0ptEs,2(A)≪d,sfalse|Afalse|2s−3+ηs,$$\begin{equation*}\hskip6pc E_{s,2}(A) \ll _{d,s} |A|^{2s -3 + \eta _{s}}, \end{equation*}$$where η3=1/2$\eta _3 = 1/2$, and ηs=(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)=x2$\psi (x) = x^2$ and s=3$s=3$, we have 72.0ptE3,2(A)≪εfalse|Afalse|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 ψfalse(xfalse)$\psi (x)$. The novelty of our estimates is that they only depend on d$d$, s$s$ and false|Afalse|$|A|$, 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.

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“…In particular, Bourgain and Demeter [2, Question 2.13] conjectured that for all finite, non‐empty sets

A

of real numbers and for all

\epsilon &gt; 0

, we have

\begin{equation} J_{3,2}(A) \ll _{\epsilon } |A|^{3 + \epsilon }. \end{equation}

Combining this with (), we see that () is equivalent to the following conjecture (see also [7, Conjecture 1.5]). Conjecture Let

s \geqslant 3

be a natural number, let

A

be a finite, non‐empty set of real numbers and let

\epsilon &gt; 0

be a real number.…”

Section: Introductionmentioning

confidence: 63%

“…These particular types of sumsets were studied in [7], and from the point of view of additive combinatorics, estimates on cardinalities of such sets are closely related to bounds for

E_{s,2}(A)

. In this paper, we record some further threshold‐breaking lower bounds for cardinalities of sumsets of the above form.…”

Section: Introductionmentioning

confidence: 99%

“…With these definitions in hand, we note that for any fixed

m \geqslant 1

, if a set

A

is well‐spaced and

m

‐dense, then () implies that

\begin{equation*} J_{s,2}(A)\ll _{s, \epsilon } X_A^{\epsilon } |A|^{2s - 3} \ll _{s, \epsilon } |A|^{2s-3+ m\epsilon }, \end{equation*}

whence, we obtain the desired conclusion by rescaling

\epsilon

appropriately. We refer the reader to [7] for a more detailed introduction to the discussion surrounding Conjecture 1.4 and estimates of the above form.…”

Section: Introductionmentioning

confidence: 99%

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Diameter‐free estimates for the quadratic Vinogradov mean value theorem

Mudgal

2022

Proceedings of London Math Soc

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“…Through the preceding paragraphs, we have seen that discrete restriction estimates deliver bounds for the corresponding additive energies in a straightforward manner (see also [12, Proof of Theorem

1.4

]). It has been noted in several works [5, 6, 10] that this type of an implication can be reversed as well.…”

Section: Further Discussion and Applications To Discrete Restriction ...mentioning

confidence: 99%

Additive energies on spheres

Mudgal

2022

Journal of London Math Soc

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In this paper, we study additive properties of finite sets of lattice points on spheres in three and four dimensions. Thus, given d,m∈double-struckN$d,m \in \mathbb {N}$, let A$A$ be a set of lattice points (x1,⋯,xd)∈Zd$(x_1, \dots , x_d) \in \mathbb {Z}^d$ satisfying x12+⋯+xd2=m$x_1^2 + \dots + x_{d}^2 = m$. When d=4$d=4$, we prove threshold breaking bounds for the additive energy of A$A$, that is, we show that there are at most Oε(mεfalse|Afalse|2+1/3−1/2766)$O_{\epsilon }(m^{\epsilon }|A|^{2 + 1/3 - 1/2766})$ solutions to the equation a1+a2=a3+a4$a_1 + a_2 = a_3 + a_4$, with a1,⋯,a4∈A$a_1, \dots , a_4 \in A$. This improves upon a result of Bourgain and Demeter, and makes progress towards one of their conjectures. A further novelty of our method is that we are able to distinguish between the case of the sphere and the paraboloid in double-struckZ4$\mathbb {Z}^4$, since the threshold bound is sharp in the latter case. We also obtain variants of this estimate when d=3$d=3$, where we improve upon previous results of Benatar and Maffucci concerning lattice point correlations. Finally, we use our bounds on additive energies to deliver discrete restriction‐type estimates for the sphere.

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A quadratic Vinogradov mean value theorem in finite fields

Mansfield,

Mudgal

2024

The Quarterly Journal of Mathematics

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Let p be a prime, let $s \geq 3$ be a natural number and let $A \subseteq \mathbb{F}_p$ be a non-empty set satisfying $|A| \ll p^{1/2}$. Denoting $J_s(A)$ to be the number of solutions to the system of equations $$ \sum_{i=1}^{s} (x_i - x_{i+s}) = \sum_{i=1}^{s} (x_i^2 - x_{i+s}^2) = 0, $$ with $x_1, \dots, x_{2s} \in A$, our main result implies that $$ J_s(A) \ll |A|^{2s - 2 - 1/9}. $$ This can be seen as a finite field analogue of the quadratic Vinogradov mean value theorem. Our techniques involve a variety of combinatorial geometric estimates, including studying incidences between Cartesian products A × A and a special family of modular hyperbolae.

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Arithmetic combinatorics on Vinogradov systems

Abstract: In this paper, we present a variant of the Balog-Szemerédi-Gowers theorem for the Vinogradov system. We then use our result to deduce a higher degree analogue of the sum-product phenomenon.

Cited by 3 publications

References 15 publications

Diameter‐free estimates for the quadratic Vinogradov mean value theorem

Diameter‐free estimates for the quadratic Vinogradov mean value theorem

Additive energies on spheres

A quadratic Vinogradov mean value theorem in finite fields

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