Related papers: On The Local Sum Conjecture in Two Dimensions
We consider the following question: Let $S_1$ and $S_2$ be two smooth, totally-real surfaces in $\mathbb{C}^2$ that contain the origin. If the union of their tangent planes is locally polynomially convex at the origin, then is $S_1 \cup…
The main objective of this paper is to prove a new inequality for plurisubharmonic functions estimating their supremum over a ball by their supremum over a measurable subset of the ball. We apply this result to study local properties of…
We give a proof of Fourier extension conjecture on the paraboloid in all dimensions bigger than 2 that begins with a decomposition suggested in Sawyer [Saw8] of writing a smooth Alpert projection as a sum of pieces whose Fourier extensions…
We combine two of Igusa's conjectures with recent semi-continuity results by Musta\c{t}\u{a} and Popa to form a new, natural conjecture about bounds for exponential sums. These bounds have a deceivingly simple and general formulation in…
The purpose of this note is to give an affirmative answer to a conjecture appearing in [Integral Transforms Spec. Funct. 26 (2015) 90-95].
We show that if an exponential sum with multiplicative coefficients is large then the associated multiplicative function is "pretentious". This leads to applications in the circle method, and a natural interpretation of the local-global…
Every graph G can be embedded in a Euclidean space as a two-distance set. This allows us to reformulate the analogue of Borsuk's conjecture for two-distance sets in terms of graphs. This conjecture remains open for dimensions from 4 to 63.…
Consider a sum of convex functions, where the only information known about each individual summand is the location of a minimizer. In this work, we give an exact characterization of the set of possible minimizers of the sum. Our results…
Explicit determinations of several classes of trigonometric sums are given. These sums can be viewed as analogues or generalizations of Gauss sums. In a previous paper, two of the present authors considered primarily sine sums associated…
In this work we prove the local multiplicity at most one theorem underlying the definition and theory of local $\gamma$-, $\epsilon$- and $L$-factors, defined by virtue of the generalized doubling method, over any local field of…
We prove the Invariant Subspace Conjecture for separable Hilbert spaces.
We give a proof of the $A_2$ conjecture in geometrically doubling metric spaces (GDMS), i.e. a metric space where one can fit not more than a fixed amount of disjoint balls of radius $r$ in a ball of radius $2r$. Our proof consists of three…
In the current paper we present a new proof of the small ball inequality in two dimensions. More importantly, this new argument, based on an approach inspired by lacunary Fourier series, reveals the first formal connection between this…
Spatial conjunction is a powerful construct for reasoning about dynamically allocated data structures, as well as concurrent, distributed and mobile computation. While researchers have identified many uses of spatial conjunction, its…
We have derived an analytical formulation for estimating the volume of geometries enclosed by implicitly defined surfaces. The novelty of this work is due to two aspects. First we provide a general analytical formulation for all…
We provide an elementary proof of a lemma that plays an important role in the classification of parallel mean curvature surfaces in two-dimensional complex space forms.
We evaluate in closed form several classes of finite trigonometric sums. Two general methods are used. The first is new and involves sums of roots of unity. The second uses contour integration and extends a previous method used by two of…
We give a proof of the Greene-Krantz conjecture on convex domains in $\CC^2$. Curiously, the proof technique depends on subelliptic estimates for the $\bar{\partial}$ problem.
The article treats some questions around Gromov's filling area conjecture. It intended to show that any filling with volume $< 2 \pi$ would not be attained and to show a local systolic inequality, implying an a priori lower estimate on the…
In this note, we present a probabilistic proof of the well-known finite geometric series. The proof follows by taking the moments of the sum and the difference of two independent exponentially distributed random variables.