R. Nair

On Riemann sums and Lebesgue integrals

AbstractCall a sequence of positive integers(mk)k=1∞ a chain ifmk devidesmk+1 and that it has dimensiond if it is a subset of the set of least common multiples ofd chains. In this paper we give a new and elementary proof that iff∈L(logL)d−1([0, 1)) and(mk)k=1∞ is of dimensiond then

On the Lebesgue measure of the expressible set of certain sequences

\mathop {\lim }\limits_{N \to \infty } \frac{1}{N}\sum\limits_{n = 1}^N {f\left( {\left\{ {x + \frac{n}{{m_N }}} \right\}} \right)} = \int_X {fd\mu , a.e.,}

Glasner sets and polynomials in primes

with respect to Lebesgue measure. This result was first proved byL. Dubins andJ. Pitman [2] using martingale theory.

On asymptotic distribution on the a-adic integers

Abstract Given a subset S of Z and a sequence I = (In)n=1∞ of intervals of increasing length contained in Z, let b ( S , I ) = lim sup ⁡ | I n | → ∞ | S ∩ I n | | I n | and let b(S) = supI b(S, I), where the supremum is taken over all sequences I = (In)n=1∞ of increasing length contained in Z. We call b(S) the Banach density of S. We will say that a sequence of natural numbers k = (κn)n=1∞ is multiply intersective, if given any subset E of the natural numbers of positive Banach density, there exists another subset R of Z with lim ⁡ N → ∞ | R ∩ [ 1 , N ] | N existing and not less than b(E), such that for each finite subset {n1, , nr} of R we have b ( E ∩ ( E + k n 1 ) ∩ ... ∩ ( E + k n r ) ) > 0. Let #x003B8; = (θn)n=1∞ be a suitable sequence of N-valued independent, idemically distributed random variables. In this paper we show that if (κn)n=1∞ is multiply intersective, then so is (kn + θn)n=1∞ for almost all θ = (θn)n=1∞

ON THE METRIC THEORY OF CONTINUED FRACTIONS IN POSITIVE CHARACTERISTIC

Abstract The paper gives a condition for the expressible set of a sequence to have Lebesgue measure zero.

Polynomial Actions in Positive Characteristic

Abstract. Let ? be a class of real valued integrable functions on [0,1). We will call a strictly increasing sequence of natural numbers an sequence if for every f in ? we have almost everywhere with respect to Lebesgue measure. Here, for a real number y we have used to denote the fractional part of y. For a finite set A we use to denote its cardinality. In this paper we show that for strictly increasing sequences of natural numbers and , both of which are sequences for all , if there exists such that then the sequence of products of pairs of elements in a and b once ordered by size is also an sequence.

Optimal continued fractions and the moving average ergodic theorem

A set of integers S is said to be Glasner if for every infinite subset A of the torus TF = R/Z and ? > 0 there exists some n E S such that the dilation nA = {nx: x E A} intersects every integral of length E in TF. In this paper we show that if Pn denotes the nth prime integer and f is any non-constant polynomial mapping the natural numbers to themselves, then (f(Pn))nzl is Glasner. The theorem is proved in a quantitative form and generalizes a result of Alon and Peres (1992).

On strong uniform distribution III

We show that the values of a polynomial with a-adic coefficients at integer and rational prime arguments are asymptotically distributed on the a-adic integers and that the integer parts of certain sequences known to be uniformly distributed modulo one, are uniformly distributed on the a-adic integers.