We would like to thank Professors Serguei Denissov and Alexander Kiselev for their constant support throughout the research and writing processes. We are grateful to them for suggesting the topic and providing us with the opportunity to apply the concepts we have learned throughout our mathematical careers. We thank them both for their excellent advice and encouragement.Abstract. For θ ∈ [0, 2π), consider the rotation matrix R θ and h = λ 0 0 0 , λ > 1.Let Wn(θ) denote the product of m R θ 's and n h's with the condition m ≤ [ǫn] (0 < ǫ < 1). We analyze the measure of the set of θ for which Wn(θ) ≥ λ δn (0 < δ < 1). This can be regarded as a model problem for the so-called Bochi-Fayad conjecture.
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