Paperfolding, Automata, and Rational Functions - Diagonals and ...

A Counter-example in Analysis
I claim that, whatever the choice of signs ±,
max
0≤θ≤1 |
n−1
h=0
X
±e 2πihθ | ≥ √ n.
Indeed, by well known orthogonality relations,
Z 1 ˛
˛Xn−1
˛ ±e 2πihθ
˛ 2
dθ = n.
0
h=0
But what is min± max0≤θ≤1 | Pn−1 h=0 ±e2πihθ |?
After suitable statistical incantations, almost surely
min max
± 0≤θ≤1 |
n−1
h=0
X
±e 2πihθ | = O( √ n log log n).
Perhaps the correct question is: What choices (i0, . . . , in−1) minimise
max
0≤θ≤1 |
n−1
h=0
X
(−1) ih 2πihθ
e | ?
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