Lower Bounds for Sums of Roots of Unity

Definitions and mathematical statement

Let ${\mathbb{N}}_{>0}=\{1,2,3,\ldots\}$. For $d\in{\mathbb{N}}_{>0}$, define the set of $d$-th roots of unity by \[ C_d=\{e^{2\pi i k/d}:k\in{\mathbb{Z}}\}\subset{\mathbb{C}}. \] For $m\in{\mathbb{N}}_{>0}$, its $m$-fold Minkowski sum is \[ mC_d=\{z_1+\cdots+z_m:z_1,\ldots,z_m\in C_d\}, \] where repetitions are allowed. For a finite set $A\subset{\mathbb{C}}$ containing a nonzero element, put \[ \min_0(A)=\min\{|z|:z\in A\setminus\{0\}\}. \] This minimum is defined for $mC_d$, since this set is finite and contains the nonzero number $m$.

The original source record asks for the following asymptotic lower bound: \[ \forall m\in{\mathbb{N}}_{>0}\ \exists E(m)\in{\mathbb{R}}_{>0}\ \exists D(m)\in{\mathbb{N}}_{>0}\ \forall d\in{\mathbb{N}}_{>0},\qquad d\ge D(m)\ \Longrightarrow\ \min_0(mC_d)>d^{-E(m)}. \] Equivalently, for each fixed $m$, every nonzero sum of $m$ roots in $C_d$ should have modulus greater than $d^{-E(m)}$ for all sufficiently large positive integers $d$, with the exponent independent of the chosen roots and of $d$.

Unresolved boundary qualification. The supplied clean statement instead asks for \[ \forall m\in{\mathbb{N}}_{>0}\ \exists E(m)\in{\mathbb{R}}_{>0}\ \forall d\in{\mathbb{N}}_{>0},\qquad \min_0(mC_d)>d^{-E(m)}. \] This differs from the original wording "as $d\to\infty$." Moreover, at $m=d=1$, the latter displayed inequality reads $1>1$. Thus the all-positive-$d$ formulation cannot be retained as an equivalent formulation without a boundary correction. The supplied evidence does not determine whether the intended correction is an eventual bound or an explicit restriction on $d$; both supplied formulations and this discrepancy require human review.

Short English statement

For each fixed number $m$ of summands, the original question asks whether every nonzero sum of $m$ $d$-th roots of unity has modulus greater than a fixed inverse power of $d$ for all sufficiently large $d$. The supplied clean version requires the inequality for every positive $d$, creating an unresolved boundary discrepancy at $m=d=1$.

Sources

• Local registry: data/unsolvedmath/problems.json, supplied record with permanent UnsolvedMath ID $30002178$, title Lower Bounds for Sums of Roots of Unity, exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2012.
• Supplied source citation: Low-Dimensional Topology and Number Theory (2013), Oberwolfach Reports, pp. 2541--2596, DOI 10.4171/owr/2012/42. The supplied literature list identifies Oberwolfach Report 42, Conjecture 7: https://doi.org/10.4171/owr/2012/42.
• Supplied literature reference: Barber, small sums of five roots of unity, https://doi.org/10.1112/blms.12826.
• Supplied literature reference: Kalogirou, small sums of roots of unity, https://arxiv.org/abs/2607.06098.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the general fixed-$m$ lower-bound problem open and reports that recent work provides stronger constructions of small sums and isolated fixed-conductor results, without establishing the conjectured uniform lower bound. This is supplied dated curation, not an independently verified literature assessment.
• The registry labels the statement corrected_verified and says that it was checked against the official report. These are supplied verification claims; the report and cited literature have not been independently consulted for this draft.