Multifractal Analysis of Bernoulli Convolutions Associated with Salem Numbers

We consider the multifractal structure of the Bernoulli convolution νλ, where λ−1 is a Salem number in (1, 2). Let τ(q) denote the L spectrum of νλ. We show that if α ∈ [τ ′(+∞), τ ′(0+)], then the level set E(α) := { x ∈ R : lim r→0 log νλ([x− r, x+ r]) log r = α } is non-empty and dimH E(α) = τ∗(α), where τ∗ denotes the Legendre transform of τ . This result extends to all self-conformal measures satisfying the asymptotically weak separation condition. We point out that the interval [τ ′(+∞), τ ′(0+)] is not a singleton when λ−1 is the largest real root of the polynomial xn−xn−1−· · ·−x+1, n ≥ 4.