**Discrete Gaussian Measures and New Bounds of the Smoothing Parameter for Lattices**

*Zhongxiang Zheng and Guangwu Xu and Chunhuan Zhao*

**Abstract: **In this paper, we start with a discussion of discrete Gaussian measures on lattices.
Several results of Banaszczyk are analyzed, different approaches are suggested.
In the second part of the paper we prove two new bounds for the smoothing parameter of lattices.
Under the natural assumption that $\varepsilon$ is suitably small, we obtain two estimations of the
smoothing parameter:

1. \[ \eta_{\varepsilon}(\mathbb{Z}) \le \sqrt{\frac{\ln \big(\frac{\varepsilon}{44}+\frac{2}{\varepsilon}\big)}{\pi}}. \]

2. For a lattice ${\cal L}\subset \mathbb{R}^n$ of dimension $n$, \[ \eta_{\varepsilon}({\cal L}) \le \sqrt{\frac{\ln \big(n-1+\frac{2n}{\varepsilon}\big)}{\pi}}\tilde{bl}({\cal L}). \]

**Category / Keywords: **Lattices, discrete Gaussian measure, smoothing parameter

**Date: **received 27 Aug 2018, last revised 29 Aug 2018

**Contact author: **gxu4uwm at uwm edu

**Available format(s): **PDF | BibTeX Citation

**Version: **20180901:024259 (All versions of this report)

**Short URL: **ia.cr/2018/786

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