On small fractional parts of perturbed polynomials

On the Fractional Parts of a/n

We give various results about the distribution of the sequence {a/n}n≥1 modulo 1, where a ≥ 2 is a fixed integer. In particular, we find and infinite subsequence A such that {a/n}n∈A is well distributed. Also we show that for any constant c > 0 and a sufficiently large N , the fractional parts of the first N terms of this sequence hit every interval J ⊆ [0, 1] of length |J | ≥ cN−0.475.

Decomposition of perturbed Chebyshev polynomials

We characterize polynomial decomposition fn = r ◦ q with r, q ∈ C[x] of perturbed Chebyshev polynomials defined by the recurrence f0(x) = b, f1(x) = x− c, fn+1(x) = (x− d)fn(x)− afn−1(x), n ≥ 1, where a, b, c, d ∈ R and a > 0. These polynomials generalize the Chebyshev polynomials, which are obtained by setting a = 1/4, c = d = 0 and b ∈ {1, 2}. At the core of the method, two algorithms for pol...

Fractional differential equations with Legendre polynomials

The application of a new mean value theorem to the fractional parts of polynomials

The application of a new mean value theorem to the fractional parts of polynomials 1. Introduction. The new mean value estimates provided by recent advances in Waring's problem (see, in particular, [9, 10, 11, 14]) may plainly be applied to improve current estimates for αn k , the distance from αn

Fractional Parts of Bernoulli Numbers

The sequence {B2}, {B4}, {B6}, ... is dense in the unit interval [0, 1], but it is not uniformly distributed [4]. Certain rational numbers appear with positive probability: 1/6 is most likely with probability 0.151..., 29/30 is next with probability 0.064... [5]. In fact, the limiting distribution F is piecewise linear with countably many jump discontinuities: F increases only when jumping (see...