## Monday, February 10, 2020

### Manish Mishra, IISER, Pune

Title: A generalization of the 3d distance theorem

Speaker: Manish Mishra, IISER Pune

Where: Seminar Room, School of Physical Sciences (SPS), C V Raman Marg, JNU

When: Tuesday, February 11, 2020, 4 PM

Abstract:

Let be a positive rational number. Call a function → to have finite gaps property mod if the following holds: for any positive irrational α and positive integer M, when the values of f(), 1 ≤ ≤ M, are inserted mod into the interval [0,P) and arranged in increasing order, the number of distinct gaps between successive terms is bounded by a constant kwhich depends only on f. In this note, we prove a generalization of the 3d distance theorem of Chung and Graham. As a consequence, we show that a piecewise linear map with rational slopes and having only finitely many non- differentiable points has finite gaps property mod P. We also show that if is distance to the nearest integer function, then it has finite gaps property mod $1$ with $k_≤ 6$. This is a joint work with Amy Philip.