Polynomial configurations in subsets of random and pseudo-random sets

Polynomial Configurations in Subsets of Random and Pseudo-random Sets

N , which are analogous to the quantitative version of the wellknown Furstenberg-Sárközy theorem due to Balog, Pintz, Pelikán, and Szemerédi. In the dense case, Balog et al showed that there is a constant C > 0 such that for all integer k 2 any subset of the first N integers of density at least C(logN) 1 4 log log log logN contains a configuration of the form {x, x+ d} for some integer d > 0. L...

On two-point configurations in subsets of pseudo-random sets

We prove a transference type result for pseudo-random subsets of Z N that is analogous to the well-known Fürstenberg-Sárközy theorem. More precisely, let k 2 be an integer and let and be real numbers satisfying + ()/(2 k+1 3) > 1. Let ✓ Z N be a set with size at least N and linear bias at most N. Then, every A ✓ with relative density |A|/|| (log log N) 1 2 log log log log log N contains a pair ...

Infinite Subsets of Random Sets of Integers

There is an infinite subset of a Martin-Löf random set of integers that does not compute any Martin-Löf random set of integers. To prove this, we show that each real of positive effective Hausdorff dimension computes an infinite subset of a Martin-Löf random set of integers, and apply a result of Miller.

Random and Pseudo-Random Structures

Propagation Models and Fitting Them for the Boolean Random Sets

In order to study the relationship between random Boolean sets and some explanatory variables, this paper introduces a Propagation model. This model can be applied when corresponding Poisson process of the Boolean model is related to explanatory variables and the random grains are not affected by these variables. An approximation for the likelihood is used to find pseudo-maximum likelihood esti...