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

We fix admissible (a_i) and let \mathsf{A}[X]:=\sum_{i\in X} a_i.

**Facts.**

if X is then \mathsf{A}[X] is not 1-random

the range of \mathsf{A} is [0, \mathsf{A}[\mathbb{N}]].

for each \beta\leq \mathsf{A}[\mathbb{N}]] there exists B\leq_T \beta with \mathsf{A}[B]=\beta.

**Question.** Is it true that:

for every computable \beta,\gamma with \beta+\gamma < \sum_i a_i there exist disjoint computable B, G such that \beta=\mathsf{A}[B] and \gamma=\mathsf{A}[G].

for every computable (r_i) with \sum_i r_i computable and \sum_i r_i < \sum_i a_i there exists computable sequence of pairwise disjoint D_i with r_i\geq \sum_{j\in D_i} a_j.

**Question.** Is it true that:

there X such that \mathsf{A}[X] is Schnorr random?

\mathsf{A}[\emptyset'] is Schnorr random?

for each left c.e. \alpha which is not Schnorr random there exists a total Solovay test (I_i) capturing \alpha and a computable injection g such that \mu(I_i)\leq a_{g(i)}?

Given total Solovay test (I_i) with \sum_i a_i> \sum_i \mu(I_i) we can effectively define pairwise disjoint intervals (J_i) with a_i>\mu(J_i) and each I_i is included in the union of finitely many J_t?

**Definition.** Let \leq_s be the Solovay reducibility on the
left c.e. reals and

\leq_s^1 be the Solovay reducibility relative to \emptyset', on the \emptyset'-left c.e. reals

x\ll_s^1 y denote \lim_n(K^{\emptyset'}(y\upharpoonright_n)-K^{\emptyset'}(x\upharpoonright_n))=\infty.

For left c.e. reals, \alpha<_s\gamma iff there exists rational c>0 and left c.e. \beta with c\cdot \gamma=\alpha+\beta.

**Exercise.** Show that there are \emptyset'-left c.e. reals \alpha,\gamma such that \alpha is random, \beta is not random and \alpha\leq_s^1 \gamma?

*Remark 1.* This says that 1-randomness is not upward \leq_s^1-closed in the \emptyset'-left c.e. reals. In contrast,
2-randomness is upward \leq_s^1-closed
in the \emptyset'-left c.e.
reals.

*Hint.* You need to produce \emptyset'-left c.e. \alpha,\gamma which are not \emptyset'-random, and although \alpha is less compressible than \gamma, relative to oracle \emptyset' it is more compressible than
\gamma. For example, \alpha=\Omega and \gamma=\Omega^{\emptyset'}\oplus
\emptyset. Here you need to use the fact that if \alpha\ll_K^1\gamma then \alpha\ll_s^1\gamma; this is a relativization
of a known fact due to Stephan.

*Remark 2.* By the hint above, there exists \beta\leq_T\emptyset' such that \Omega+\beta is not random. By the result in
`differences of halting probabilitiesâ€™, this \beta can be chosen to be random and a
d.c.e.Â 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. \alpha there exists random
\beta such that \alpha-\beta is not random.

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

non-random left c.e. real \beta such that \alpha+\beta is Schnorr random and nonrandom?

left c.e. real \beta such that \alpha+\beta is not Schnorr random?

The question if, given left c.e. \alpha there exists left c.e. \beta with \alpha+\beta having a certain property is the same as asking if there exists left c.e. \gamma\geq_s \alpha with the said property.

So we ask if for each nonrandom left c.e. real \alpha there exists:

left c.e. real \gamma\geq_s \alpha which is nonrandom and Schnorr random ?

left c.e. real \gamma\geq_s \alpha 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 \mu(V)=1/2 and admissible sequences (a_i), (b_i), two players pick sets F_0, G_0, F_1, G_1,\dots in turns, so that

player 1 picks pairwise disjoint finite F_0, F_1,\dots

player 2 picks pairwise disjoint finite G_0, G_1, \dots

r_{2s}:=\sum_{t < s} \sum_{i\in F_t} a_{i}+\sum_{t < s} \sum_{i\in G_t} b_{i} is defined after 2s moves

r_{2s+1}:=\sum_{t\leq s} \sum_{i\in F_t} a_{i}+\sum_{t < s} \sum_{i\in G_t} b_{i} is defined after 2s+1 moves.

Player 2 wins if there exists increasing (t_s) with \forall s,\ r_{2s}\not\in V_{t_s}. Who wins?