Global maximum of ?(x)-x

[Gene Smith]

On 6/11/06, Gerald McGarvey <Gerald.McGarvey at comcast.net> wrote:
>
> Based on a laborious semi-manual process (not recommended) of
> narrowing down the maximum, I believe the maximum of ?(x)-x
> occurs at a constant c whose continued fraction begins
> [0; 1, 3, 1, 4, 1, 4, 1, 5, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 5, 1, 4, 1,
> 4,
> 1, 4, 1, 4, 1, 4, 1, 5, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 5, 1, 4, 1, 4, 1,
> 4, 1, 4, 1, 4, 1, 5, 1, 4, 1, 4, 1, 4, 1, 4, 1, ...


I think the process would be more expeditious if you used the  Conway box
function instead, which is easy to compute exactly for dyadic
rationals--meaning N/2^m type numbers.
The question of the maximum of ?(x)-x is the same as the
question of the maximum of x-box(x), where box, the Conway box function, is
the inverse ? function. If x is a level m dyadic rational, that is, N/2^m,
then I think by comparing
f(x) = x-box(x) for the values x, x+2^(-m-1), x-2^(-m-1) you should be able
to determine the maximum more easily.


>
> Has it be proven that ?(x) + ?(1-x) = 1 for all x?



Yes.
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