We investigate the box dimensions of the horizon of a fractal surface defined by a function f C[0, 1]~2. In particular we show that a prevalent surface satisfies the 'horizon property', namely that the box dimension of the horizon is one less than that of the surface. Since a prevalent surface has box dimension 3, this does not give us any information about the horizon of surfaces of dimension strictly less than 3. To examine this situation we introduce spaces of functions with surfaces of upper box dimension at most a, for a α [2, 3). In this setting the behaviour of the horizon is more subtle. We construct a prevalent subset of these spaces where the lower box dimension of the horizon lies between the dimension of the surface minus one and 2. We show that in the sense of prevalence these bounds are as tight as possible if the spaces are defined purely in terms of dimension. However, if we work in Lipschitz spaces, the horizon property does indeed hold for prevalent functions. Along the way, we obtain a range of properties of box dimensions of sums of functions.
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机译：我们研究了由函数f C [0，1]〜2定义的分形表面的水平线的框尺寸。特别是，我们证明了流行的表面满足“水平属性”，即水平线的框尺寸比表面的框尺寸小一。由于一个流行的曲面的框尺寸为3，因此不会提供任何关于尺寸小于3的曲面的水平线的信息。为了检查这种情况，我们引入了函数空间，其中最大框尺寸为a的曲面最大为a。 [2，3）。在这种设置下，地平线的行为更加微妙。我们构造了这些空间的一个普遍子集，其中水平线的下框尺寸介于曲面的尺寸减去1和2之间。我们证明，如果仅在空间中定义空间，则在普遍意义上，这些边界尽可能紧密尺寸方面。但是，如果我们在Lipschitz空间中工作，那么horizontal属性确实适用于流行函数。在此过程中，我们获得了函数总和的盒子尺寸的一系列属性。
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