{"paper":{"title":"Coefficients of $q$-real numbers: their combinatorial meaning and growth","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.QA"],"primary_cat":"math.CO","authors_text":"Pavel Etingof, Valentin Ovsienko","submitted_at":"2026-08-07T03:28:01Z","abstract_excerpt":"A $q$-deformed real number, or ``$q$-real'', was defined by Morier-Genoud and the second author. When $x\\in\\mathbb{R}$ such that $x\\geq0$, the $q$-analogue $[x]_q$ is a power series with integer coefficients in one formal variable~$q$. In general a $q$-real is a formal Laurent series. The main goal of this paper is to study the coefficients of $q$-reals as functions on~$\\mathbb{R}$ and give a combinatorial interpretation of these coefficients. This allows us to prove a conjecture studied by several authors stating that the $q$-deformed golden ratio has the smallest radius of convergence among "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.06761","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2608.06761/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}