Some Remarks on a Lemma of Ran Raz

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Abstract.

In this note we will review a Lemma published by Ran Raz in [R], and suggest improvements and extensions. Raz’ Lemma compares the measure of a set on the sphere to the measure of its section with a random subspace. Essentially, it is a sampling argument. It shows that, in some sense, we can simultaneously sample a function on the entire sphere and in a random subspace.

In the first section we will discuss some preliminary ideas, which underlie the lemma and our interest in it. We will view a random subspace as the span of random points, without discussing the sampling inside the subspace. We will demonstrate how substantial results follow from this elementary approach. In the second section we will review the original proof of Raz’ Lemma, analyse it, and improve the result. In the final section we will extend the Lemma to other settings.