In this paper, we make a novel connection between information theory and additive combinatorics; more specifically, between deletion/insertion correcting codes and the celebrated Littlewood–Offord problem. We see how results from one area can have an impact on the other area and vice versa. In particular, a result on the Littlewood–Offord problem gives a nice upper bound for the size of the Levenshtein code and the Helberg code (and possibly other variants of these codes). Also, a recent result on the deletion correcting codes gives a modular analogue of the Littlewood–Offord problem which generalizes the results of Vaughan and Wooley (Q J Math Oxf Ser (2) 42(2):379–386, 1991) (obtained using tools from analytic number theory and properties of exponential sums) and of Griggs (Bull Am Math Soc (N.S.) 28:329–333, 1993) (obtained using a combinatorial argument). This novel connection might opens up new doors to research in these or other related areas.

中文翻译：

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删除校正码满足 Littlewood–Offord 问题

在本文中，我们在信息论和加性组合学之间建立了一种新的联系；更具体地说，在删除/插入纠正代码和著名的 Littlewood-Offord 问题之间。我们看到一个领域的结果如何影响另一个领域，反之亦然。特别是，Littlewood-Offord 问题的结果为 Levenshtein 代码和 Helberg 代码（以及这些代码的其他可能变体）的大小提供了一个很好的上限。此外，最近关于删除校正码的结果给出了 Littlewood–Offord 问题的模模拟，该问题概括了 Vaughan 和 Wooley (QJ Math Oxf Ser (2) 42(2):379–386, 1991)（使用获得来自解析数论和指数和性质的工具）和 Griggs（Bull Am Math Soc (NS) 28:329–333，1993）（使用组合论证获得）。这种新颖的联系可能会为这些或其他相关领域的研究打开新的大门。