On Zaremba's Conjecture

For $b\in\mathbb{N}, b\ge2$ we determine the limit points of certain subsets of $$ \left\{\frac{b^n\pmod{n}}{n}:n\in\mathbb{N}\right\}. $$ As a consequence, we obtain the density of the latter set in $[0,1]$, a result first established in 2013 by Cilleruelo, Kumchev, Luca, Ru\'{e} and Shparlinski..

Every positive rational number has representations as Egyptian fractions (sums of reciprocals of distinct positive integers) with arbitrarily many terms and with arbitrarily large denominators. However, such representations normally use a very sparse subset of the positive integers up to the largest demoninator. We show that for every positive rational there exist Egyptian fractions whose largest denominator is at most N and whose denominators form a positive proportion of th...

In this paper I present a new method of studying the densities of the Collatz trajectories generated by a set $S \subset \mathbb{N}$. This method is used to furnish an alternative proof that $d(\{y \in \mathbb{N} : \exists k \text{ where } T^k(y) < cy\}) = 1$ for all $c > 0$. Finally, I briefly discuss how the ideas presented in this paper could be used to improve the result that $d(\{y \in \mathbb{N} : \exists k \text{ where } T^k(y) < y^c\}) = 1$ for $c > \log_4 3$.

In this paper we give a detailed measure theoretical analysis of what we call sum-level sets for regular continued fraction expansions. The first main result is to settle a recent conjecture of Fiala and Kleban, which asserts that the Lebesgue measure of these level sets decays to zero, for the level tending to infinity. The second and third main result then give precise asymptotic estimates for this decay. The proofs of these results are based on recent progress in infinite ...

We ask, for which $n$ does there exists a $k$, $1 \leq k < n$ and $(k,n)=1$, so that $k/n$ has a continued fraction whose partial quotients are bounded in average by a constant $B$? This question is intimately connected with several other well-known problems, and we provide a lower bound in the case of B=2.

The Erdos-Davenport theorem on the multiples claims that for any set of natural numbers the set consisting of their multiples possesses the logarithmic density. An analogous statement is proved for the sets of rational multiples.

Introducing the notion of a rational system of measure preserving transformations and proving a recurrence result for such systems, we give sufficient conditions in order a subset of rational numbers to contain arbitrary long arithmetic progressions.

A partition of the positive integers into sets $A$ and $B$ {\em avoids} a set $S\subset\N$ if no two distinct elements in the same part have a sum in $S$. If the partition is unique, $S$ is {\em uniquely avoidable.} For any irrational $\alpha>1$, Chow and Long constructed a partition which avoids the numerators of all convergents to $\alpha$, and conjectured that the set $S_\alpha$ which this partition avoided was uniquely avoidable. We prove that the set of numerators of con...

We give a short proof of a sumset conjecture of Erd\"os, recently proved by Moreira, Richter and Robertson: every subset of the integers of positive density contains the sum of two infinite sets. The proof is written in the framework of classical ergodic theory.

Nathanson constructed asymptotic bases for the integers with a prescribed representation function, then asked how dense they can be. We can easily obtain an upper bound using a simple argument. In this paper, we will see this is indeed the best bound we can get for asymptotic bases for the integers with an arbitrary representation function prescribed.