{"paper":{"title":"The discrepancy of $(n_kx)$ with respect to certain probability measures","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Agamemnon Zafeiropoulos, Niclas Technau","submitted_at":"2018-12-15T14:18:38Z","abstract_excerpt":"Let $(n_k)_{k=1}^{\\infty}$ be a lacunary sequence of integers. We show that if $\\mu$ is a probability measure on $[0,1)$ such that $|\\widehat{\\mu}(t)|\\leq c|t|^{-\\eta}$, then for $\\mu$-almost all $x$, the discrepancy $D_N(n_kx)$ satisfies \\begin{equation*} \\frac{1}{4} \\leq \\limsup_{N\\to\\infty}\\frac{N D_N(n_kx)}{\\sqrt{N\\log\\log N}} \\leq C \\end{equation*} for some constant $C>0$, proving a conjecture of Haynes, Jensen and Kristensen. This allows a slight improvement on their previous result on products of the form $q\\|q\\alpha\\| \\|q\\beta-\\gamma\\| $."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1812.06293","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}