On a lemma of Littlewood and Offord
On a lemma of Littlewood and Offord
复制标题
关于利特伍德和奥福德的引理
DOI：
10.1007/978-3-642-00856-6_22
复制
发表时间：
2004
影响因子：
4.9
通讯作者：
G. Ziegler
中科院分区：
文献类型：
--
作者：
M. Aigner;G. Ziegler
In their work on the distribution of roots of algebraic equations, Littlewood and Offord proved in 1943 the following result: Let a 1, a 2, . . . , a n be complex numbers with |a i | ≥ 1 for all i, and consider the 2 n linear combinations
$$ \sum^{n}_{i=1} \varepsilon_{i} a_{i} $$
with e i ∈ {1,−1}. Then the number of sums \( \sum^{n}_{i=1} \varepsilon_{i} a_{i} \) which lie in the interior of any circle of radius 1 is not greater than
$$ c \frac{2^{n}}{\sqrt{n}} \log n $$
, for some constant c > 0.
In their work on the distribution of roots of algebraic equations, Littlewood and Offord proved in 1943 the following result: Let a 1, a 2, . . . , a n be complex numbers with |a i | ≥ 1 for all i, and consider the 2 n linear combinations
$$ \sum^{n}_{i=1} \varepsilon_{i} a_{i} $$
with e i ∈ {1,−1}. Then the number of sums \( \sum^{n}_{i=1} \varepsilon_{i} a_{i} \) which lie in the interior of any circle of radius 1 is not greater than
$$ c \frac{2^{n}}{\sqrt{n}} \log n $$
, for some constant c > 0.