A remark on the Birkhoff ergodic theorem

Applications of the Birkhoff Ergodic Theorem

Ergodic theory studies the long-term averaging properties of measurepreserving dynamical systems. In this paper, we state and present a proof of the ergodic theorem due to George Birkhoff, who observed the asymptotic equivalence of the time-average and space-average of a point x in a finite measure space. Then, we examine a number of applications of this theorem in number-theoretic problems, in...

A Remark on a Remark by Macaulay or Enhancing Lazard Structural Theorem

A Remark on Zoloterav’s Theorem

Let n ≥ 3 be an odd integer. For any integer a prime to n, define the permutation γ a,n of {1,. .. , (n − 1)/2} by γ a,n (x) = n − {ax} n if {ax} n ≥ (n + 1)/2, {ax} n if {ax} n ≤ (n − 1)/2, where {x} n denotes the least nonnegative residue of x modulo n. In this note, we show that the sign of γ a,n coincides with the Jacobi symbol a n if n ≡ 1 (mod 4), and 1 if n ≡ 3 (mod 4).

A remark on the Erdös-Szekeres theorem

The Erdős–Szekeres theorem states that for any , any sufficiently large set of points in general position in the plane contains points in convex position. We show that for any and any finite sequence ! ! #" , with #$% & ( ')(+*, ! ! ), any sufficiently large set of points in general position in the plane contains either an empty convex ) gon or convex polygons . ! ! /. " , where 0 . $ 0,(& $ an...

A Remark on the Gromov Convergence Theorem

In [3] M. Gromov introduced the concept of convergence of Riemannian manifolds and he proved the convergence theorem. Since that time the theorem has been developed in detail (see [5], [7], [2]), and we know that it contains some interesting applications. Nevertheless there seems to be an inadequate way of applying the convergence theorem. The purpose of this paper is to present an example whic...