Perfect fractal sets with zero Fourier dimension and arbitrarily long   arithmetic progressions
  By considering a Moran-type construction of fractals on $[0,1]$, we show that for any $0\le s\le 1$, there exists some Moran fractal set, which is perfect, with Hausdorff dimension $s$ whose Fourier dimension is zero and it contains arbitrarily long arithmetic progressions. 
Chun-Kit Lai
math.CA