I'm reading through the paper:

https://arxiv.org/abs/1802.04411

However, I'm having a difficult time understanding the following proof: Lemma 2.1 page 3).

I understand the general goal of the proof, but the second half in which we establish a lower bound for the integral contains several steps that I find puzzling. Where (and how) do we make use of the fact that $\eta=c_1^{0.1}$, and why was this specific value chosen? Where does the $\sqrt{3/4}$ come from? Why exactly does $0<\beta\leq 2$ imply that the smallness of $c_1$ is independent of $\beta$? Any help would be greatly appreciated.

I'm reading through the paper  Poincaré type and spectral gap inequalities with fractional Laplacians on Hamming cube.

However, I'm having a difficult time understanding the following proof: Lemma 2.1 page 3).

I understand the general goal of the proof, but the second half in which we establish a lower bound for the integral contains several steps that I find puzzling. Where (and how) do we make use of the fact that $\eta=c_1^{0.1}$, and why was this specific value chosen? Where does the $\sqrt{3/4}$ come from? Why exactly does $0<\beta\leq 2$ imply that the smallness of $c_1$ is independent of $\beta$? Any help would be greatly appreciated.

I'm reading through the paper:

https://arxiv.org/pdf/1802.04411.pdf

However, I'm having a difficult time understanding the following proof (page 3):

 https://i.sstatic.net/epHpF.png

I understand the general goal of the proof, but the second half in which we establish a lower bound for the integral contains several steps that I find puzzling. Where (and how) do we make use of the fact that $\eta=c_1^{0.1}$, and why was this specific value chosen? Where does the $\sqrt{3/4}$ come from? Why exactly does $0<\beta\leq 2$ imply that the smallness of $c_1$ is independent of $\beta$? Any help would be greatly appreciated.

Thanks.

I'm reading through the paper:

https://arxiv.org/abs/1802.04411

However, I'm having a difficult time understanding the following proof:  Lemma 2.1 page 3).

I understand the general goal of the proof, but the second half in which we establish a lower bound for the integral contains several steps that I find puzzling. Where (and how) do we make use of the fact that $\eta=c_1^{0.1}$, and why was this specific value chosen? Where does the $\sqrt{3/4}$ come from? Why exactly does $0<\beta\leq 2$ imply that the smallness of $c_1$ is independent of $\beta$? Any help would be greatly appreciated.

I'm reading through the paper:

https://arxiv.org/pdf/1802.04411.pdf

However, I'm having a difficult time understanding the following proof (page 3):

https://i.sstatic.net/epHpF.png

I understand the general goal of the proof, but the second half in which we establish a lower bound for the integral contains several steps that I find puzzling. Where (and how) do we make use of the fact that $\eta=c_1^{0.1}$? Where does the $\sqrt{3/4}$ come from? Why exactly does $0<\beta\leq 2$ imply that the smallness of $c_1$ is independent of $\beta$? Any help would be greatly appreciated.

Thanks.

I'm reading through the paper:

https://arxiv.org/pdf/1802.04411.pdf

However, I'm having a difficult time understanding the following proof (page 3):

https://i.sstatic.net/epHpF.png

I understand the general goal of the proof, but the second half in which we establish a lower bound for the integral contains several steps that I find puzzling. Where (and how) do we make use of the fact that $\eta=c_1^{0.1}$, and why was this specific value chosen? Where does the $\sqrt{3/4}$ come from? Why exactly does $0<\beta\leq 2$ imply that the smallness of $c_1$ is independent of $\beta$? Any help would be greatly appreciated.

Thanks.