Leírás
We call a (Erdos) distinct covering system, a set of
arithmetic progressions $\mathcal{C}={(a_i \mod m_i)}$, with 1
<m_0<m_1<...<, such that every integer belongs in at least one of the
arithmetic progressions in $\mathcal{C}$. Addressing the case of a
finite covering system with $0\leq i\leq k$, first, we review the old
result of Mirsky and Newman, which shows that
1/m_0+\cdots+1/m_k>1.
We go over joint work with Michael Filaseta, where we show that if
m_0>4, then the right hand side of the above can be replaced with 1+c,
for c an absolute positive constant. This fully answers a question of
Erdos and Selfridge and uses the distortion method which has been
applied to resolve other long-standing conjectures in the field. Time
permitting, I will at the end pose an open problem, after I present
some aspects of the #C being infinite case.
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