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Box Counting Dimension of the Union L of Trinomial Curves When α ≥ 1

Authors: Kaoutar Lamrini Uahabi, Mohamed Atounti


In the present work, we consider one category of curves denoted by L(p, k, r, n). These curves are continuous arcs which are trajectories of roots of the trinomial equation zn = αzk + (1 − α), where z is a complex number, n and k are two integers such that 1 ≤ k ≤ n − 1 and α is a real parameter greater than 1. Denoting by L the union of all trinomial curves L(p, k, r, n) and using the box counting dimension as fractal dimension, we will prove that the dimension of L is equal to 3/2.

Keywords: fractal dimension, feasible angles, trinomial curves, trinomial equation, Minkowski sausage

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