Conical inverse mapping theorems

Tame Nonsmooth Inverse Mapping Theorems

We give several versions of local and global inverse mapping theorem for tame non necessarily smooth, mappings. Here tame mapping means a mapping which is subanalytic or, more generally, definable in some o-minimal structure. Our sufficient conditions are formulated in terms of various properties (convexity, positivity of some principal minors, contractiblity) of the space of Jacobi’s matrices ...

Conical Plurisubharmonic Measure and New Cross Theorems

We study the boundary behavior of the conical plurisubharmonic measure. As an application, we establish some new extension theorems for separately holomorphic mappings defined on cross sets.

Inverse Approximation Theorems for . . .

Inverse Theorems for Subset Sums

Let A be a nite set of integers. For h 1, let S h (A) denote the set of all sums of h distinct elements of A. Let S(A) denote the set of all nonempty sums of distinct elements of A. The direct problem for subset sums is to nd lower bounds for jS h (A)j and jS(A)j in terms of jAj. The inverse problem for subset sums is to determine the structure of the extremal sets A of integers for which jS h ...

Optimal Inverse Littlewood-offord Theorems

Let ηi, i = 1, . . . , n be iid Bernoulli random variables, taking values ±1 with probability 1 2 . Given a multiset V of n integers v1, . . . , vn, we define the concentration probability as ρ(V ) := sup x P(v1η1 + . . . vnηn = x). A classical result of Littlewood-Offord and Erdős from the 1940s asserts that, if the vi are non-zero, then ρ(V ) is O(n−1/2). Since then, many researchers have obt...