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<p>We investigate the metric properties of products of consecutive digits in Engel expansions. More precisely, for a nonnegative real number <i>β</i> and positive integer <i>m</i>, we study the set of points in (0<i>,</i> 1] for which the normalized product of <i>m</i> successive Engel digits converges to <i>β</i>. The exact Hausdorff dimension of this set is determined. This result extends the classical metric theory of Engel series and contributes to the dimension theory of digit expansions.</p>

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Metric properties of digit products in Engel expansions

  • 期刊论文

  • 发表日期：

    2026年4月29日

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  • Jingjing Wu,
  • Yan Feng

摘要

We investigate the metric properties of products of consecutive digits in Engel expansions. More precisely, for a nonnegative real number β and positive integer m, we study the set of points in (0, 1] for which the normalized product of m successive Engel digits converges to β. The exact Hausdorff dimension of this set is determined. This result extends the classical metric theory of Engel series and contributes to the dimension theory of digit expansions.