Composite of Randomness and Hindman-Theorem Questions

Definitions and mathematical statement

The supplied record concatenates two independent questions. Its statement status is unrecoverable; the following records both questions under explicit conventional interpretations, without promoting them to a verified clean statement.

1. A real $x\in[0,1)$ is Martin--Löf random for Lebesgue measure if, for every uniformly effectively open sequence $(U_k)_{k\geq1}$ with $\lambda(U_k)\leq 2^{-k}$, one has $x\notin\bigcap_{k\geq1}U_k$. Here effectively open means a computably enumerable union of intervals with rational endpoints, relative to $[0,1)$.

   For $t\in{\mathbb{R}}$, let $\{t\}=t-\lfloor t\rfloor$. Set $y_j=\{2^j x\}$, for $j\geq1$, and define the star discrepancy of the first $N$ terms by \[ D_N^*(x)=\sup_{0\leq t\leq1}\left|\frac{1}{N}\#\{1\leq j\leq N:y_j<t\}-t\right|. \] Under this interpretation, the first question asks whether there exist a Martin--Löf random $x\in[0,1)$, a constant $C>0$, and an integer $N_0\geq2$ such that \[ \forall N\geq N_0,\qquad D_N^*(x)\leq C\frac{\log N}{N}. \] The source does not specify the domain and randomness measure, reduction modulo one, or the discrepancy convention. The definitions above make the usual unit-interval interpretation explicit. If discrepancy instead means the supremum over all half-open subintervals $[a,b)\subseteq[0,1)$, its value $D_N$ satisfies $D_N^*\leq D_N\leq2D_N^*$, so that choice gives the same existence question with this asymptotic bound. Other intended conventions require source clarification.

2. For this reconstruction let ${\mathbb{N}}=\{0,1,2,\ldots\}$. For an integer $n\geq1$ and an infinite set $H\subseteq{\mathbb{N}}$, define \[ \operatorname{FS}_n(H)=\{h_1+\cdots+h_n:h_1,\ldots,h_n\in H,\ h_1<\cdots<h_n\}. \] An instance of the exact-length, two-color restriction of Hindman's theorem is a total coloring $c:{\mathbb{N}}\to\{0,1\}$. A solution is an infinite $H\subseteq{\mathbb{N}}$ for which $c$ is constant on $\operatorname{FS}_n(H)$. An instance is computable when $c$ is computable. A set $H$ is $\Sigma^0_n$ when its membership predicate is definable by an unrelativized arithmetical $\Sigma^0_n$ formula, with $n$ alternating blocks of number quantifiers beginning existentially and a computable matrix; unused quantifier blocks are permitted.

   Under these conventions, the second question asks for which integers $n\geq1$ the following holds: \[ \begin{gathered} \exists\text{ total computable }c:{\mathbb{N}}\to\{0,1\}\quad \forall H\subseteq{\mathbb{N}},\\ \left[ H\text{ is infinite}\ \land\ \exists i\in\{0,1\}\ \forall s\in\operatorname{FS}_n(H),\ c(s)=i \right] \ \Longrightarrow\ H\notin\Sigma^0_n. \end{gathered} \] Here sums use exactly $n$ distinct elements, rather than allowing repeated summands or all lengths at most $n$. The source does not explicitly specify these conventions, whether natural numbers include zero, or the representation of a solution; these are stated interpretations requiring review.

The supplied dated curation labels the composite record partially solved and reports a computable exact-two-sum, two-color instance with no $\Sigma^0_2$ solution. It describes the general $n$ question and the Martin--Löf randomness question as unresolved. These are supplied literature assessments, not independently verified mathematical or current-status claims.

Short English statement

This composite record asks two independent questions: whether a Martin--Löf random real can have discrepancy bounded by $C\log N/N$ along its doubling orbit, and which exact sum lengths $n$ admit a computable two-coloring with no infinite $\Sigma^0_n$ homogeneous solution. The displayed formulations state conventional interpretations explicitly; the source marks the composite statement unrecoverable, and human mathematical review is required.

Sources

• Supplied local registry record: permanent UnsolvedMath ID $30003690$, exact source problem code catalogue entry, title "Composite of Randomness and Hindman-Theorem Questions," in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2018.
• Supplied report citation: Computability Theory (2019), Oberwolfach Reports, DOI 10.4171/owr/2018/1, pp. 5--41. https://doi.org/10.4171/owr/2018/1. The background attributes the questions to workshop open-problem discussions documented in Oberwolfach Reports. The report title's year and the DOI's year component are retained as supplied.
• Supplied literature reference: "Csima et al., exact-two-sum Hindman theorem." https://math.uchicago.edu/~drh/Papers/hindman.html. The record supplies no fuller author list or bibliographic details.
• Supplied literature reference: "Oberwolfach report." https://doi.org/10.4171/owr/2018/1.
• Literature assessment dated exactly 2026-08-21: supplied dated curation, not independently verified. It labels the record partially solved, reports the exact-two-sum two-color result with no $\Sigma^0_2$ solution, and describes the general $n$ and Martin--Löf discrepancy questions as unresolved. The background expressly requests independent verification of time-sensitive status and cited claims.
• Supplied statement-verification note: the official open-problem section is reported to show that the extraction concatenates independent questions; no separate clean statement is promoted, and the original remains visible for provenance. This verification account is supplied metadata, not a claim that the sources were visited for this draft.