... 1) a F3(k+1).Singularity of Fractal Measure Associated with 172.3. If y3k+3= 7. By similar argument as above cases, we get #t3k+3 a F3(k+1).Thus, by using inductive hypothesis and ... convolution associated with the goldennumber, Trans. Amer. Math. Soc. 349(1997), 2917 - 2940.5. T. Hu and N. Nguyen, Local dimensions of fractal probability measures associated with equal probability ... Fractal, Wavelets, East - WestJ. of Math. Vol2,No1(2000), 55-71.8. J. C. Lagarias and Y. Wang, Tiling the line with translates of one tile, InventionsMath. 124(1996), 341 - 365.9. S. M. Ngai...