Conditions for the Difference Set of a Central Cantor Set to be a Cantorval

Abstract Let $$C(\lambda )\subset [0,1]$$ C ( ? ) ? [ 0 , 1 ] denote the central Cantor set generated by a sequence $$\lambda =(\lambda _{n})\in \left( 0,\frac{1}{2} \right) ^{\mathbb {N}}$$ = n ? 2 N . By known trichotomy, difference $$ C(\lambda )-C(\lambda )$$ - of is one three possible sets: finite union closed intervals, set, or Cantorval. Our main result describes effective conditions for $$(\lambda _{n})$$ which guarantee that We show these can be expressed in several equivalent forms. Under additional assumptions, measure Cantorval established. give an application proved theorems achievement sets some fast convergent series.