Let
n
be an arbitrary integer, let
p
be a prime factor of
n
. Denote by
ω1
the
pth
primitive unity root, \documentclass{aastex}
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$$\omega _1 : = e^{\tfrac{{2\pi i}}
{p}}$$
\end{document}.Define
ωi
≔
ω1i
for 0 ≦
i
≦
p
− 1 and
B
≔ {1,
ω1
, …,
ωp
−1
}
n
⊆ ℂ
n
.Denote by
K
(
n; p
) the minimum
k
for which there exist vectors
ν1
, …,
νk
∈
B
such that for any vector
w
∈
B
, there is an
i
, 1 ≦
i
≦
k
, such that
νi
·
w
= 0, where
ν · w
is the usual scalar product of
ν
and
w
.Gröbner basis methods and linear algebra proof gives the lower bound
K
(
n; p
) ≧
n
(
p
− 1).Galvin posed the following problem: Let
m
=
m
(
n
) denote the minimal integer such that there exists subsets
A1
, …,
Am
of {1, …, 4
n
} with |
Ai
| = 2
n
for each 1 ≦
i
≦
n
, such that for any subset
B
⊆ [4
n
] with 2
n
elements there is at least one
i
, 1 ≦
i
≦
m
, with
Ai
∩
B
having
n
elements. We obtain here the result
m
(
p
) ≧
p
in the case of
p
> 3 primes.
NON-HERMITIAN OSCILLATOR-LIKE HAMILTONIANS AND λ-COHERENT STATES REVISITED
