Speaker: Siddharth Iyer
Affiliation: University of New South Wales
Title: Distribution modulo 1
Date and Time: November 13, 2024, 14:30:00 Hours
Venue: A-369
Asbtract: We make two contributions in the study of fractional parts. 1) It is shown that rationals with denominators made of digits $0$ and $1$ in base $b$ well approximate real numbers. 2) We improve upon a result by Steinerberger (2024) by showing that for $\alpha \in \mathbb{R}$, there exists integers $1 \leq b_{1},...,b_{k} \leq n$ such that: $\left\| \sum_{j=1}^{k}\sqrt{b_{j}} -\alpha\right\| = O(n^{- \gamma_{k}})$, where $\gamma_{k} \geq \frac{k-1}{4}$ and $\gamma_{k} = k/2$ when $k = 2^{m-1}$, $m = 1,2,\ldots$.
Affiliation: University of New South Wales
Title: Distribution modulo 1
Date and Time: November 13, 2024, 14:30:00 Hours
Venue: A-369
Asbtract: We make two contributions in the study of fractional parts. 1) It is shown that rationals with denominators made of digits $0$ and $1$ in base $b$ well approximate real numbers. 2) We improve upon a result by Steinerberger (2024) by showing that for $\alpha \in \mathbb{R}$, there exists integers $1 \leq b_{1},...,b_{k} \leq n$ such that: $\left\| \sum_{j=1}^{k}\sqrt{b_{j}} -\alpha\right\| = O(n^{- \gamma_{k}})$, where $\gamma_{k} \geq \frac{k-1}{4}$ and $\gamma_{k} = k/2$ when $k = 2^{m-1}$, $m = 1,2,\ldots$.