Maximum Additive Multiplicity in Logarithmic Random Sets

Definitions and mathematical statement

Let ${\mathbb{N}}_{>0}=\{1,2,\ldots\}$. Let $(\xi_i)_{i\geq1}$ be independent Bernoulli random variables satisfying \[ \mathbb P(\xi_i=1)=\frac1i,\qquad \mathbb P(\xi_i=0)=1-\frac1i, \] and define the logarithmic random set \[ A=\{i\in{\mathbb{N}}_{>0}:\xi_i=1\}. \] In particular, $1\in A$ almost surely. The positive-integer indexing is required for the supplied probabilities $1/i$ to be defined.

The supplied problem asks to determine \[ \max_x r_A(x), \] where $r_A(x)$ counts representations of $x$ as a sum of elements of $A$.

Unresolved formulation. The source record does not specify whether summands must be distinct, whether repetitions are allowed, whether order matters, or whether the number of summands is fixed or arbitrary. For example, under the additional convention that representations are finite subsets of distinct elements and that $x$ is a positive integer, the representation function would be \[ r_A^{\mathrm{dist}}(x)=\#\left\{B\subseteq A: B\text{ is finite and }\sum_{b\in B}b=x\right\},\qquad x\in{\mathbb{N}}_{>0}. \] This is only a possible interpretation, not a convention established by the supplied record. The domain of $x$, the treatment of the empty sum if $x=0$ is admitted, and the intended probabilistic meaning of "determine" are also unspecified. In addition, the source writes a maximum over an unrestricted domain without specifying whether it is attained or whether an extended supremum is intended. Consequently, the supplied evidence does not determine a unique precise mathematical target. No truncation, growth parameter, or almost-sure assertion is added here.

Short English statement

Independently include each positive integer $i$ in $A$ with probability $1/i$, and determine the largest number of representations of a number as a sum of elements of $A$. The representation convention, range of the represented number, interpretation of the unrestricted maximum, and intended probabilistic conclusion require clarification.

Sources

• Local supplied UnsolvedMath registry record, permanent ID $30004293$, source problem code catalogue entry, titled "Maximum Additive Multiplicity in Logarithmic Random Sets." No individual proposer is supplied; the registry gives proposed year 2019.
• Supplied report citation: Analytic Number Theory (2020), Oberwolfach Reports, DOI 10.4171/owr/2019/50, pp. 3141--3205. This is also the supplied literature source labeled "original OWR report": https://doi.org/10.4171/owr/2019/50.
• Supplied literature source: Ford--Green--Koukoulopoulos, "equal sums in random sets": https://arxiv.org/abs/1908.00378. The supplied attribution names the authors of this literature source, not an identified proposer of the problem.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and states that the unrestricted maximum is not determined, while the main paper studies truncated variants and obtains partial lower bounds or variational characterizations. This is supplied dated curation, not an independently verified literature assessment. The supplied background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified, but its accompanying explanation limits the repairs to formula delimiters, notation, grammar, and typography, with mathematical content and scope preserved. This label does not establish mathematical verification of the formulation above.