We study the depletion of a diffusing substance [described by a scalar Laplacian field u(r)] in the vicinity of an absorbing fractal, consisting of a Gaussian random walk (RW) or a self-avoiding walk (SAW) in d dimensions. The moments 〈[u(r${)]}^{n}$〉 of the field u(r) at a distance r from a point on an absorber of linear size R obey 〈[u(r${)]}^{n}$〉\ensuremath{\sim}(r/R${)}^{\ensuremath{\lambda}(n)}$, where the \ensuremath{\lambda}(n) are a set of independent exponents. The incident flux \ensuremath{\varphi} of the field u defines a fractal measure on the absorber. The scaling dimensions D(n) describing this measure obey (1-n)[D(n)-D]=n\ensuremath{\lambda}(1)-\ensuremath{\lambda}(n), with D the fractal dimension; moreover, \ensuremath{\lambda}(1)=D+2-d. These relations apply for any fractal absorber. For the RW and SAW, we observe that the exponents \ensuremath{\lambda}(n) are the same as those describing the statistics of linear chains and star polymers with selective excluded-volume interactions. This allows us to calculate \ensuremath{\lambda}(n) and D(n) explicitly to order ${\ensuremath{\epsilon}}^{2}$, where \ensuremath{\epsilon}=4-d. We find D(n)=D-n${\ensuremath{\epsilon}}^{2}$/4 for the RW and D(n)=D-9n${\ensuremath{\epsilon}}^{2}$/64 for the SAW. We also study nonperturbatively the limit of high n. For n large, D(n)-D(\ensuremath{\infty})\ensuremath{\sim}${n}^{(3\mathrm{\ensuremath{-}}d)/(d\mathrm{\ensuremath{-}}2)}$ for 3ldl4, whereas in d=3 dimensions, D(n)-D(\ensuremath{\infty})\ensuremath{\sim}[ln(n${)]}^{\mathrm{\ensuremath{-}}1}$; here D(\ensuremath{\infty})=d-2. These high-n results apply to the RW and SAW alike. We construct for the absorbing RW part of the scaling function f(\ensuremath{\alpha}), defined by Halsey et al. [Phys. Rev. A 33, 1141 (1986)], and find a range of \ensuremath{\alpha} for which f(\ensuremath{\alpha}) is negative. Identifying f(\ensuremath{\alpha}) with the histogram of the measure expressed in logarithmic variables, we discuss the meaning of negative f(\ensuremath{\alpha}).