Exponential Small-Value Bounds for Littlewood Polynomials

Definitions and mathematical statement

For each integer $d\geq 0$, define \[ \mathcal P_d=\left\{\sum_{j=0}^{d}a_jx^j\in{\mathbb{Z}}[x]:a_j\in\{-1,0,1\}\text{ for }0\leq j\leq d\right\}. \] Thus leading coefficients may vanish, and the zero polynomial belongs to $\mathcal P_d$. The symbol $\#S$ denotes the cardinality of a finite set $S$.

A precise candidate reading of the supplied question is the following: does there exist a constant $C>0$, independent of the rational number and of $d$, such that, for every pair of coprime integers $p,q$ with $q>0$ and \[ \frac{9}{10}\leq\frac pq\leq\frac{19}{20}, \] there exists a constant $c_{p,q}>0$, independent of $d$, for which \[ \forall d\in{\mathbb{Z}}_{\geq 1},\qquad \#\left\{P\in\mathcal P_d:\left|P\!\left(\frac pq\right)\right|<c_{p,q}e^{-Cd}\right\}<e^{d/100}? \] Here both inequalities defining the small-value count and its upper bound are strict, and both endpoints of the rational interval are included.

Unresolved qualifications. The displayed quantifier order interprets "absolute" as requiring one $C$ for all rational numbers in the interval. The source places the existential phrase for $C$ after the rational quantifier, so this uniformity requires confirmation. Positivity of $c_{p,q}$, its independence of $d$, the range $d\geq1$, and the reduced-fraction convention are explicit interpretations rather than fully stated source hypotheses. If $c_{p,q}>0$ and $d=0$ is included, the zero polynomial makes the strict bound impossible, since its right-hand side is $1$. The supplied background also describes this record as a conditional application of Question 4, but supplies neither that question nor the relevant conditional implication; no such implication can be stated precisely from this record alone.

Short English statement

The question asks whether one absolute exponential scale ensures that, at every rational point in $[9/10,19/20]$, fewer than $e^{d/100}$ polynomials with coefficients in $\{-1,0,1\}$ have extremely small values, with a positive prefactor allowed to depend on the rational point. The displayed formulation requires confirmation of the degree range, quantifier order, and missing connection to Question 4. This is a statement-only draft requiring human mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003114$, exact source problem code catalogue entry, title "Exponential Small-Value Bounds for Littlewood Polynomials," in the Oberwolfach Reports Open Problems collection. No individual proposer is identified; the supplied proposed year is 2016.
• Supplied report citation: Diophantische Approximationen (2017), Oberwolfach Reports, pp. 1099--1187, DOI 10.4171/owr/2016/21: https://doi.org/10.4171/owr/2016/21. The record describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source, "OWR report": https://ems.press/content/serial-article-files/46625?nt=1.
• Supplied literature source, "Varju": https://arxiv.org/abs/1608.04210. No fuller bibliographic description or precise supporting result is supplied.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the record open and describes it as a conditional application of the still-open Question 4, rather than a separately resolved theorem. This is supplied dated curation, not independently verified; the background expressly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and says its mathematical content was preserved while notation, typography, or omitted definitions were repaired. These are supplied editorial labels, not independent mathematical verification of this draft.