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Consider the following matrix $A$:

$A = begin{bmatrix}l_1F & cdots & l_mF\I_N & cdots & I_Nend{bmatrix}$.

Here, ${l_1, cdots, l_m}$ are distinct real numbers such that their sum is zero, $I_N$ is the identity matrix of dimension $Ntimes N$ and $F$ is a $rtimes N$ matrix given by:
$ F = begin{bmatrix}
frac{1}{N} & frac{1}{N}e^{frac{2pi k_1}{N}} &cdots & frac{1}{N}e^{frac{2pi k_1 (N-1)}{N}}\
vdots & vdots & vdots & vdots \
frac{1}{N} & frac{1}{N}e^{frac{2pi k_r}{N}} &cdots & frac{1}{N}e^{frac{2pi k_r (N-1)}{N}}
end{bmatrix},
$
where ${k_1, cdots, k_r}$ are distinct integers such that $1leq k_1, cdots, k_r leq N/100$ (ofcourse, assuming $N$ is much larger than 100.).

Question: What is a non-trivial lower bound (as a function of ${l_1, cdots, l_m}$, ${k_1, cdots, k_r}$, $N$ and $r$) for the smallest singular value of any submatrix made up of $N+r$ linearly independent columns of $A$?

It is also clear that as $N$ becomes larger and the other parameters remain constant, the smallest singular value will go to zero. It would be equally good if one can show that the rate of decrease is $Oleft(frac{1}{N^{1.5-delta}}right)$, $delta>0$.

An observation might be of some help - the rows of $A$ are orthogonal.