<p>I hadn't heard of this before either, so  I am little defensive, since I <a href="https://arxiv.org/abs/1603.09596" rel="nofollow noreferrer">have seen that real and synthetic datasets in high dimensions</a> really do not support the claim of the paper in question.</p>

<p>As a result, what I would suggest, as a first, dirty, clumsy and maybe not good first attempt is to generate a sphere in a dimension of your choice (I <a href="https://gsamaras.wordpress.com/code/create-pointset-on-a-sphere-or-a-klein-bottle/" rel="nofollow noreferrer">do it like like this</a>) and then place a query at the center of the sphere. </p>

<p>In that case, every point lies in the same distance with the query point, thus the Nearest Neighbor has a distance equal to the Farthest Neighbor.</p>

<p>This, of course, is independent from the dimension, but it's what came at a thought after looking at the figures of the paper. It should be enough to get you stared, but surely, better datasets may be generated, if any.</p>

<hr>

<p>Edit about:</p>

<blockquote>
  <p>distances for each point got bigger with more dimensions!!!!</p>
</blockquote>

<p>this is expected, since the higher the dimensional space, the sparser the space is, thus the greater the distance is. Moreover, this is expected, if you think for example, the Euclidean distance, which gets grater as the dimensions grow.</p>
Dataset for meaningless “Nearest Neighbor”?
<p>In the paper "<a href="https://cis.temple.edu/~vasilis/Courses/CIS750/Papers/beyer99when_17.pdf" rel="nofollow noreferrer">When Is 'Nearest Neighbor' Meaningful?</a>" we read that, "We show that under certain broad conditions (in terms of data and query distributions, or workload), as dimensionality increases, the distance to the nearest
neighbor approaches the distance to the farthest neighbor. In other words, the contrast in distances to different data points becomes nonexistent. The conditions
we have identified in which this happens are much broader than the independent and identically distributed (IID) dimensions assumption that other work
assumes."</p>

<p>My question is how I should generate a dataset that resembles this effect? I have created three points each with 1000 dimensions with random numbers ranging from 0-255 for each dimension but points create different distances and do not reproduce what is mentioned above. It seems changing dimensions (e.g. 10 or 100 or 1000 dimensions) and ranges (e.g. [0,1]) do not change anything. I still get different distances which should not be any problem for e.g. clustering algorithms! </p>