According to a theorem of Szemerédi (1975), for every positive integer k and every δ > 0, there exists S = S(k, δ) such that every subset X of {1, 2, . . . , S} of size at least δS contains an arithmetic progression with k terms. Here we generalize the above result to any set of real numbers. For every positive integer k and every c > 0, there exists N = N(k, c) such that every set of n ≥ N poi...

A theorem of Green, Tao, and Ziegler can be stated as follows: if R is a pseudorandom distribution, and D is a dense distribution of R, then D can be modeled as a distribution M which is dense in uniform distribution such that D and M are indistinguishable. The reduction involved in the proof has exponential loss in the distinguishing probability. Reingold et al give a new proof of the theorem ...

In this paper we give very sharp bounds for the distance between regularity and strong regularity. The solution of this problem uses a general matrix analogue [18] of the Perron-Frobenius Theory. This theory has been graded as a challenge problem for future research by the International Linear Algebra Society [7]. An interval matrix [A] ∈ IIMn(IR) is called regular, if every A ∈ [A] is nonsingu...

In this paper we will generalize  some of known results on the tight closure of an ideal to the tight closure of an ideal relative to a module .

Abstract: The purpose of this paper is to discuss the properties of regularity and strong regularity of fuzzy measure on metric spaces following the previous results. Some properties are defined with the help of nulladditivity such as inner\outer regularity and the regularity of fuzzy measure. We define the strong regularity of fuzzy measures and show our result that the null-additive fuzzy mea...

We show that Souslin Absoluteness and Projective Regularity holds ii Souslin Uniformization does. As a result, Souslin Absoluteness plus 1 n Projective Regularity implies 1 n+1 Projective Regularity. Another result is that 1 5 Souslin Absoluteness implies 1 4 Projective Regularity, and 1 6 Souslin Absoluteness implies 1 5 Projective Regularity.

We devise a framework which leads to the formulation of a unified theory of normality (regularity), semi-normality (semi-regularity), s-normality (s-regularity), feeblynormality (feebly-regularity), pre-normality (pre-regularity), and others. Certain aspects of theory are given by unified proof.