**Definition.** We say that a rational sequence (*a*_{i}) is {} if it
is computable, *a*_{i} < ∑_{j > i}*a*_{j}
and ∑_{j > i}*a*_{j}
is a computable real < 1.

We fix admissible (*a*_{i}) and let
A[*X*] := ∑_{i ∈ X}*a*_{i}.

**Facts.**

if

*X*is then A[*X*] is not 1-randomthe range of A is [0,A[ℕ]].

for each

*β*≤ A[ℕ]] there exists*B*≤_{T}*β*with A[*B*] =*β*.

**Question.** Is it true that:

for every computable

*β*,*γ*with*β*+*γ*< ∑_{i}*a*_{i}there exist disjoint computable*B*,*G*such that*β*= A[*B*] and*γ*= A[*G*].for every computable (

*r*_{i}) with ∑_{i}*r*_{i}computable and ∑_{i}*r*_{i}< ∑_{i}*a*_{i}there exists computable sequence of pairwise disjoint*D*_{i}with*r*_{i}≥ ∑_{j ∈ Di}*a*_{j}.

**Question.** Is it true that:

there

*X*such that A[*X*] is Schnorr random?A[∅′] is Schnorr random?

for each left c.e.

*α*which is not Schnorr random there exists a total Solovay test (*I*_{i}) capturing*α*and a computable injection*g*such that*μ*(*I*_{i}) ≤*a*_{g(i)}?Given total Solovay test (

*I*_{i}) with ∑_{i}*a*_{i}> ∑_{i}*μ*(*I*_{i}) we can effectively define pairwise disjoint intervals (*J*_{i}) with*a*_{i}>*μ*(*J*_{i}) and each*I*_{i}is included in the union of finitely many*J*_{t}?

**Definition.** Let ≤_{s} be the Solovay
reducibility on the left c.e. reals and

≤

_{s}^{1}be the Solovay reducibility relative to ∅′, on the ∅′-left c.e. reals*x*≪_{s}^{1}*y*denote lim_{n}(*K*^{∅′}(*y*↾_{n})−*K*^{∅′}(*x*↾_{n})) = ∞.

For left c.e. reals, *α*<_{s}*γ*
iff there exists rational *c* > 0 and left c.e. *β* with *c* ⋅ *γ* = *α* + *β*.

**Exercise.** Show that there are ∅′-left c.e. reals *α*, *γ* such that *α* is random, *β* is not random and *α*≤_{s}^{1}*γ*?

*Remark 1.* This says that 1-randomness is not upward ≤_{s}^{1}-closed in
the ∅′-left c.e. reals. In contrast,
2-randomness is upward ≤_{s}^{1}-closed in
the ∅′-left c.e. reals.

*Hint.* You need to produce ∅′-left c.e. *α*, *γ* which are not ∅′-random, and although *α* is less compressible than *γ*, relative to oracle ∅′ it is more compressible than *γ*. For example, *α* = *Ω* and *γ* = *Ω*^{∅′} ⊕ ∅.
Here you need to use the fact that if *α*≪_{K}^{1}*γ*
then *α*≪_{s}^{1}*γ*;
this is a relativization of a known fact due to Stephan.

*Remark 2.* By the hint above, there exists *β*≤_{T}∅′ such that
*Ω* + *β* is not random.
By the result in `differences of halting probabilities’, this *β* can be chosen to be random and a
d.c.e. real (the difference of two left c.e. reals). So the sum of two
(positive) random reals may not be random. In this case, it is necessary
that one of them is left c.e. and the other is a real. For each random
left c.e. *α* there exists
random *β* such that *α* − *β* is not random.

**Question.** Is it true that for each nonrandom left
c.e. real *α* there exists:

non-random left c.e. real

*β*such that*α*+*β*is Schnorr random and nonrandom?left c.e. real

*β*such that*α*+*β*is not Schnorr random?

The question if, given left c.e. *α* there exists left c.e. *β* with *α* + *β* having a certain
property is the same as asking if there exists left c.e. *γ*≥_{s}*α*
with the said property.

So we ask if for each nonrandom left c.e. real *α* there exists:

left c.e. real

*γ*≥_{s}*α*which is nonrandom and Schnorr random ?left c.e. real

*γ*≥_{s}*α*which is not Schnorr random ?

Two players produce an increasing sequence *r*_{s} by adding
increases in turn. Player 2 wins if *r* := lim_{s}*r*_{s}
is outside a given open set *V*.
The increases available to Player 1 are chosen (without repetition) from
the terms of a given admissible (*a*_{i}) (many *a*_{i} can be chosen
at once). Similarly for Player 2 with respect to admissible (*b*_{i}).

Formally, given an effectively open set *V* with *μ*(*V*) = 1/2 and admissible
sequences (*a*_{i}), (*b*_{i}),
two players pick sets *F*_{0}, *G*_{0}, *F*_{1}, *G*_{1}, …
in turns, so that

player 1 picks pairwise disjoint finite

*F*_{0},*F*_{1}, …player 2 picks pairwise disjoint finite

*G*_{0},*G*_{1}, …*r*_{2s}:= ∑_{t < s}∑_{i ∈ Ft}*a*_{i}+ ∑_{t < s}∑_{i ∈ Gt}*b*_{i}is defined after 2*s*moves*r*_{2s + 1}:= ∑_{t ≤ s}∑_{i ∈ Ft}*a*_{i}+ ∑_{t < s}∑_{i ∈ Gt}*b*_{i}is defined after 2*s*+ 1 moves.

Player 2 wins if there exists increasing (*t*_{s}) with ∀*s*, *r*_{2s} ∉ *V*_{ts}.
Who wins?