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Let    q  ∈  (  1  ,  2  )   . A   q  ‐expansion of a number   x   in    [  0  ,  1  /  (  q  ‐  1  )  ]    is a sequence(   δ  i   )    i  =  1   ∞   ∈    {  0  ,  1  }   Nsatisfying    x  =    ∑   i  =  1   ∞δ  iq  i.Let    B   ℵ  0denote the set of   q   for which there exists   x   with a countable number of   q  ‐expansions, and let    B   1  ,   ℵ  0denote the set of   q   for which 1 has a countable number of   q  ‐expansions. In Erdős  et al  [On the uniqueness of the expansions    1  =   ∑   i  =  1   ∞    q   ‐   n  i.  Acta Math. Hungar.   58  (1991), 333–342] it was shown that    min   B   ℵ  0    =  min   B   1  ,   ℵ  0=   (  1  +   5   )   /  2   , and in S. Baker [On small bases which admit countably many expansions.  J. Number Theory   147  (2015), 515–532] it was shown thatB   ℵ  0    ∩   (   (  1  +   5   )   /  2  ,   q  1   ]   =   {   q  1   }    , whereq  1(  ≈  1  .  64541  )is the positive root ofx  6   ‐   x  4   ‐   x  3   ‐  2   x  2   ‐  x  ‐  1  =  0   . In this paper we show that the second smallest point of    B   1  ,   ℵ  0isq  3(  ≈  1  .  68042  )    , the positive root ofx  5   ‐   x  4   ‐   x  3   ‐  x  +  1  =  0   . En route to proving this result, we show thatB   ℵ  0    ∩   (   q  1   ,   q  3   ]   =   {   q  2   ,   q  3   }    , whereq  2(  ≈  1  .  65462  )is the positive root ofx  6   ‐  2   x  4   ‐   x  3   ‐  1  =  0   .

ON SMALL BASES FOR WHICH 1 HAS COUNTABLY MANY EXPANSIONS