  It is well established that the notion of min-entropy fails to satisfy the
\emph{chain rule} of the form $H(X,Y) = H(X|Y)+H(Y)$, known for Shannon
Entropy. Such a property would help to analyze how min-entropy is split among
smaller blocks. Problems of this kind arise for example when constructing
extractors and dispersers.
  We show that any sequence of variables exhibits a very strong strong
block-source structure (conditional distributions of blocks are nearly flat)
when we \emph{spoil few correlated bits}. This implies, conditioned on the
spoiled bits, that \emph{splitting-recombination properties} hold. In
particular, we have many nice properties that min-entropy doesn't obey in
general, for example strong chain rules, "information can't hurt" inequalities,
equivalences of average and worst-case conditional entropy definitions and
others. Quantitatively, for any sequence $X_1,\ldots,X_t$ of random variables
over an alphabet $\mathcal{X}$ we prove that, when conditioned on $m = t\cdot
O( \log\log|\mathcal{X}| + \log\log(1/\epsilon) + \log t)$ bits of auxiliary
information, all conditional distributions of the form $X_i|X_{<i}$ are
$\epsilon$-close to be nearly flat (only a constant factor away). The argument
is combinatorial (based on simplex coverings).
  This result may be used as a generic tool for \emph{exhibiting block-source
structures}. We demonstrate this by reproving the fundamental converter due to
Nisan and Zuckermann (\emph{J. Computer and System Sciences, 1996}), which
shows that sampling blocks from a min-entropy source roughly preserves the
entropy rate. Our bound implies, only by straightforward chain rules, an
additive loss of $o(1)$ (for sufficiently many samples), which qualitatively
meets the first tighter analysis of this problem due to Vadhan
(\emph{CRYPTO'03}), obtained by large deviation techniques.
