A reinforcement of the Bourgain-Kontorovich's theorem

Let $F \subseteq [0,1]$ be a set that supports a probability measure $\mu$ with the property that $ |\widehat{\mu}(t)| \ll (\log |t|)^{-A}$ for some constant $ A > 0 $. Let $\mathcal{A}= (q_n)_{n\in \mathbb{N}} $ be a sequence of natural numbers. If $\mathcal{A}$ is lacunary and $A >2$, we establish a quantitative inhomogeneous Khintchine-type theorem in which (i) the points of interest are restricted to $F$ and (ii) the denominators of the `shifted' rationals are restricted ...

In 1970, Erd\H{o}s and S\'ark\"ozy wrote a joint paper studying sequences of integers $a_1<a_2<\dots$ having what they called property P, meaning that no $a_i$ divides the sum of two larger $a_j,a_k$. In the paper, it was stated that the authors believed, but could not prove, that a subset $A\subset[n]$ with property P has cardinality at most $|A|\leqslant \left\lfloor \frac{n}{3}\right\rfloor+1$. In 1997, Erd\H{o}s offered \$100 for a proof or disproof of the claim that $|A|...

We employ infinite ergodic theory to show that the even Stern-Brocot sequence and the Farey sequence are uniformly distributed mod 1 with respect to certain canonical weightings. As a corollary we derive the precise asymptotic for the Lebesgue measure of continued fraction sum-level sets as well as connections to asymptotic behaviours of geometrically and arithmetically restricted Poincar\'e series. Moreover, we give relations of our main results to elementary observations fo...

Our purpose is to give an account of the $r$-tuple problem on the increasing sequence of reduced fractions having denominators bounded by a certain size and belonging to a fixed real interval. We show that when the size grows to infinity, the proportion of the $r$-tuples of consecutive denominators with components in certain apriori fixed arithmetic progressions with the same ratio approaches a limit, which is independent on the interval. The limit is given explicitly and it ...

We extend two results about the ordinary continued fraction expansion to best simultaneous Diophantine approximations of vectors or matrices. The first is Levy-Khintchin Theorem about the almost sure growth rate of the denominators of the convergents. The second is a Theorem of Bosma, Hendrik and Wiedijk about the almost sure limit distribution of the sequence of products $q_n d(q_n\theta, Z)$ where the $q_n$'s are the denominators of the convergents associated with the real ...

We give a heuristic for the number of reduced rationals on Cantor's middle thirds set, with a fixed bound on the denominator. We also describe extensive numerical computations supporting this heuristic.

For any given real number $\alpha$ with bounded partial quotients, we construct explicitly continuum many real numbers $\beta$ with bounded partial quotients for which the pair $(\alpha, \beta)$ satisfies a strong form of the Littlewood conjecture. Our proof is elementary and rests on the basic theory of continued fractions.

In this paper we present a convergence theorem for continued fractions of the form $K_{n=1}^{\infty}a_{n}/1$. By deriving conditions on the $a_{n}$ which ensure that the odd and even parts of $K_{n=1}^{\infty}a_{n}/1$ converge, these same conditions also ensure that they converge to the same limit. Examples will be given.

Given an irrational number $\alpha$ consider its irrationality measure function $\psi_{\alpha}(t)=\min\limits_{1\le q\le t, q\in\mathbb{Z}}\|q\alpha\|$. The set of all values of $\lambda(\alpha)=(\limsup\limits_{t\to\infty} t\psi_{\alpha}(t))^{-1}$ where $\alpha $ runs through the set $\mathbb{R}\setminus\mathbb{Q}$ is called the Lagrange spectrum $\mathbb{L}$. In a paper by Moshchevitin an irrationality measure function $\psi^{[2]}_{\alpha}(t)=\min\limits_{1\le q\le t, q\in\...

As the conclusion of a line of investigation undertaken in two previous papers, we compute asymptotic frequencies for the values taken by numerators of differences of consecutive Farey fractions with denominators restricted to lie in arithmetic progression.