It seems the exact formula given by David Bar Moshe in the predecessor to this question,

What is the volume of a $\delta$-ball in the orthogonal group $O(n)$? Is there a (simple) lower bound?

is the place to start. (To cut and paste: this formula is

Vol(delta-ball) = 2^(m^2)/(pi^m * m!)* int_ phi_ 1<=phi_ 2 <= . . . <=phi_ m <=2 * arcsin(sqrt(delta/2))*

prod_ 1 < = j < k < = m (cos(phi_ k)-cos(phi_ j)^2 prod_ l sin^2(phi_ l) dphi_ 1 . . . dphi_ k.

when n=2m+1, and something very similar for n=2m (my guess is that any lower bound for the former can basically be achieved for the latter).

I am tempted to start dyadically decomposing based on the approximate size of phi_l and of phi_k-phi_j, but this is likely to lose some exponential factors of n.

An alternative is Fourier decomposition; the formula above does suggest that there is a reasonably nice formula for the integral of exp( i theta r ) where r is the distance to the identity and theta is a parameter. If one gets sufficiently good control on that integral, one can then try Fourier inversion.

It seems the exact formula given by David Bar Moshe in the predecessor to this question,

What is the volume of a $\delta$-ball in the orthogonal group $O(n)$? Is there a (simple) lower bound?

is the place to start. (To cut and paste: this formula is

Vol(delta-ball) = 2^(m^2)/(pi^m * m!)* int_ phi_ 1<=phi_ 2 <= . . . <=phi_ m <=2 * arcsin(sqrt(delta/2))*

prod_ 1 < = j < k < = m (cos(phi_ k)-cos(phi_ j)^2 prod_ l sin^2(phi_ l) dphi_ 1 . . . dphi_ k.

when n=2m+1, and something very similar for n=2m (my guess is that any lower bound for the former can basically be achieved for the latter).

I am tempted to start dyadically decomposing based on the approximate size of phi_l and of phi_k-phi_j, but this is likely to lose some exponential factors of n.

An alternative is Fourier decomposition; the formula above does suggest that there is a reasonably nice formula for the integral of exp( i theta r ) where r is the distance to the identity and theta is a parameter. If one gets sufficiently good control on that integral, one can then try Fourier inversion.

It seems the exact formula given in the predecessor to this question,

What is the volume of a $\delta$-ball in the orthogonal group $O(n)$? Is there a (simple) lower bound?

is the place to start. (To cut and paste: this formula is

Vol(delta-ball) = 2^(m^2)/(pi^m * m!)* int_ phi_ 1<=phi_ 2 <= . . . <=phi_ m <=2 * arcsin(sqrt(delta/2))*

prod_ 1 < = j < k < = m (cos(phi_ k)-cos(phi_ j)^2 prod_ l sin^2(phi_ l) dphi_ 1 . . . dphi_ k.

when n=2m+1, and something very similar for n=2m (my guess is that any lower bound for the former can basically be achieved for the latter).

I am tempted to start dyadically decomposing based on the approximate size of phi_l and of phi_k-phi_j, but this is likely to lose some exponential factors of n.

An alternative is Fourier decomposition; the formula above does suggest that there is a reasonably nice formula for the integral of exp( i theta r ) where r is the distance to the identity and theta is a parameter. If one gets sufficiently good control on that integral, one can then try Fourier inversion.

It seems the exact formula given by David Bar Moshe in the predecessor to this question,

What is the volume of a $\delta$-ball in the orthogonal group $O(n)$? Is there a (simple) lower bound?

is the place to start. (To cut and paste: this formula is

Vol(delta-ball) = 2^(m^2)/(pi^m * m!)* int_ phi_ 1<=phi_ 2 <= . . . <=phi_ m <=2 * arcsin(sqrt(delta/2))*

prod_ 1 < = j < k < = m (cos(phi_ k)-cos(phi_ j)^2 prod_ l sin^2(phi_ l) dphi_ 1 . . . dphi_ k.

when n=2m+1, and something very similar for n=2m (my guess is that any lower bound for the former can basically be achieved for the latter).

I am tempted to start dyadically decomposing based on the approximate size of phi_l and of phi_k-phi_j, but this is likely to lose some exponential factors of n.

An alternative is Fourier decomposition; the formula above does suggest that there is a reasonably nice formula for the integral of exp( i theta r ) where r is the distance to the identity and theta is a parameter. If one gets sufficiently good control on that integral, one can then try Fourier inversion.