Mark Lewko
YOU?
Author Swipe
View article: A bilinear approach to the finite field restriction problem
A bilinear approach to the finite field restriction problem Open
Let $P$ denote the $3$-dimensional paraboloid over a finite field of odd characteristic in which $-1$ is not a square. We show that the Fourier extension operator associated with $P$ maps $L^2$ to $L^{r}$ for $r > \frac{24}{7} \approx3.428…
View article: Counting rectangles and an improved restriction estimate for the paraboloid in $F_p^3$
Counting rectangles and an improved restriction estimate for the paraboloid in $F_p^3$ Open
Given $A \subset F_{p}^2$ a sufficiently small set in the plane over a prime residue field, we prove that there are at most $O_ε(|A|^{\frac{99}{41}+ε})$ rectangles with corners in $A$. The exponent $\frac{99}{41} = 2.413\ldots$ improves sl…
View article: Finite field restriction estimates based on Kakeya maximal operator estimates
Finite field restriction estimates based on Kakeya maximal operator estimates Open
In the finite field setting, we show that the restriction conjecture associated to any one of a large family of d=2n+1 dimensional quadratic surfaces implies the n+1 -dimensional Kakeya conjecture (Dvir's theorem). This includes the case o…
View article: Sidonicity and variants of Kaczmarz’s problem
Sidonicity and variants of Kaczmarz’s problem Open
We prove that a uniformly bounded system of orthonormal functions satisfying the condition: (1) must contain a Sidon subsystem of proportional size, (2) must satisfy the Rademacher–Sidon property, and (3) must have its five-fold tensor …
View article: Additive Energy and the Hausdorff dimension of the exceptional set in\n metric pair correlation problems
Additive Energy and the Hausdorff dimension of the exceptional set in\n metric pair correlation problems Open
For a sequence of integers $\\{a(x)\\}_{x \\geq 1}$ we show that the\ndistribution of the pair correlations of the fractional parts of $\\{ \\langle\n\\alpha a(x) \\rangle \\}_{x \\geq 1}$ is asymptotically Poissonian for almost all\n$\\al…
View article: The square variation of rearranged Fourier series
The square variation of rearranged Fourier series Open
We prove that there exists a rearrangement of the first $N$ elements of the trigonometric system such that the $L^2$-norm of the square variation operator is at most $O_{\epsilon}(\log^{9/22+\epsilon}(N))$. This is an improvement over $O(\…
View article: An improved lower bound related to the Furstenberg-Sárközy theorem
An improved lower bound related to the Furstenberg-Sárközy theorem Open
Let $D(n)$ denote the cardinality of the largest subset of the set $\{1,2,\ldots,n\}$ such that the difference of no pair of distinct elements is a square. A well-known theorem of Furstenberg and Sárközy states that $D(n)=o(n)$. In the oth…