 Advances in Pure Mathematics, 2011, 1, 274-275 doi:10.4236/apm.2011.15048 Published Online September 2011 (http://www.SciRP.org/journal/apm) Copyright © 2011 SciRes. APM Besicovitch-Eggleston Function Manav Das Department of Mat hematics, University of Louisville, Louisville, USA E-mail: manav@louisville.edu Received June 2, 2011; revised July 2, 2011; accepted July 15, 20 11 Abstract In this work we introduce a function based on the well-known Besicovitch-Eggleston sets, and prove that the Hausdorff dimension of its graph is 2. Keywords: Hausdorff Dimension, Multifractal, Binomial Measure, Dyadic Intervals 1. Introduction Let 0,1x, and let denote its binary expansion. For any 12=.,, =0 or 1ixxx x0< <12p we may consider the set 12=0,1:limsup npnxx xKxnp Besicovitch  prov ed that  log1log 1dim =log2Hppp ppK  where dimHA denotes the Hausdorff dimension of the set A. This result was generalized to the -ary case by Eggleston . Billing sley proved a more general ver-sion of this result in the context of probability spaces . Billingsley’s result was related to densities in , and a similar result involving packing dimensions was proved in . Sets such as NpK are studied in the context of multifractal theory (see [1,7,9,11,14-16]) and Billing- sley-type results have been proved by several authors in this context. Recently, such a result has been proved for a countable symbol space in . In this paper, we are interested in a natural function that may be defined using Besicovitch’s result. We call this the Besicovitch-Eggleston function: let 1,,= #Nnx of ’s in the first digits of the dyadic expansion for 1nx. Define 1, ,ifit existslim=1othenNnxfx nrwise This function allows us to visualize the multifractal components of 0,1 as lev el sets. If we let p denote the invariant multifractal meas ure on pK then it is clear that d=p.fyy>0p In the next section, we state and prove our main result, and finally we close with some open problems. 2. Main Result We will need the following result by Besicovitch and Moran. This is not the form in which it was originally stated and proved. However, this modern version may be found in . Theorem 1. For any there exists a constant s such that for all Borel sets we must have 0,1sb2ER1dsssyEb Ey where E=:,yExxy. We are now ready to state the main result: Theorem 2. Let =0,1Graph fB= 2HB0,1. Then and dim2Bp=0. Proof. The upper bound is obvious while the lower bound follows from Theorem 1. Fix and 0,1schoose such that log logqq>.log 2pps Therefore 1=fps. We can choose an interval sI containing such that for every p=syBsyI. It follows from Theorem 1 that . Observ-ing that was arbitrary gives us the lower bound. Moreover, since every vertical line meets exactly once, Fubini’s theorem tells us that in fact =BB1s<1s=02B. 3. Concluding Remarks Here we pose some problems related to the Besicovitch- Eggleston function. L. D. NIEM ET AL.275 1) Can one find the precise scaling function  such that ? Is the set 0<