We study the $Δ^♯$–representation of numbers belonging to $(0;1]: \sum_k (-1)^{k-1} 2^{1-a_1-a_2-...-a_k} ≡ Δ_{a_1a_2...a_n...}^♯.$ This is an encoding of numbers by means of infinite alphabet $A = \{1,2,...\}.$  Its applications in the theory of fractal dimension, fractal geometry, metric and probabilistic theory of numbers are proposed.

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