proofs

On a lemma
of Littlewood and Offord
Chapter 22
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 ∑ n
i=1 ε ia i with ε i ∈ {1, −1}.
Then the number of sums ∑ n
i=1 ε ia i which lie in the interior of any
circle of radius 1 is not greater than
c 2n
√ n
log n for some constant c > 0.
A few years later Paul Erdős improved this bound by removing the log n
term, but what is more interesting, he showed that this is, in fact, a simple
consequence of the theorem of Sperner (see page 179).
To get a feeling for his argument, let us look at the case when all a i are
real. We may assume that all a i are positive (by changing a i to −a i and ε i
to −ε i whenever a i < 0). Now suppose that a set of combinations ∑ ε i a i
lies in the interior of an interval of length 2. Let N = {1, 2, . . ., n} be the
index set. For every ∑ ε i a i we set I := {i ∈ N : ε i = 1}. Now if I I ′
for two such sets, then we conclude that
∑
ε
′
i a i − ∑ ε i a i = 2 ∑
a i ≥ 2,
i∈I ′ \I
which is a contradiction. Hence the sets I form an antichain, and we
conclude from the theorem of Sperner that there are at most ( n
⌊n/2⌋)
such
combinations. By Stirling’s formula (see page 11) we have
( ) n
⌊n/2⌋
≤
c 2n
√ n
for some c > 0.
John E. Littlewood
Sperner’s theorem. Any antichain of
subsets of an n-set has size at most
`
n
⌊n/2⌋´.
For n even and all a i = 1 we obtain ( n
n/2)
combinations
∑ n
i=1 ε ia i that
sum to 0. Looking at the interval (−1, 1) we thus find that the binomial
number gives the exact bound.
In the same paper Erdős conjectured that ( n
⌊n/2⌋)
was the right bound for
complex numbers as well (he could only prove c 2 n n −1/2 for some c) and
indeed that the same bound is valid for vectors a 1 , . . .,a n with |a i | ≥ 1 in
a real Hilbert space, when the circle of radius 1 is replaced by an open ball
of radius 1.