AN EXACT AFFINE-CORE FORMULA FOR BOUNDED UNIONS OF TRANSLATES

Abstract For a nonempty subset B of a group G , let μ G , B ( n , c ) $\mu _{G,B}(n,c)$ mu Subscript upper G comma upper B Baseline left parenthesis n comma c right parenthesis be the least possible value of max ∅ ≠ S ⊆ B , | S | ≤ c | A S | $\max _{\varnothing \ne S\subseteq B,\ |S|\le c}|AS|$ max Underscript empty set not equals upper S subset of or equal to upper B comma StartAbsoluteValue upper S EndAbsoluteValue less than or equals c Endscripts StartAbsoluteValue upper A upper S EndAbsoluteValue over all n -element subsets A ⊆ G $A\subseteq G$ upper A subset of or equal to upper G . Given b 0 ∈ B $b_0\in B$ b 0 element of upper B , set D = b 0 − 1 B $D=b_0^{-1}B$ upper D equals b 0 Superscript negative 1 Baseline upper B and H = ⟨ D ⟩ $H=\langle D\rangle $ upper H equals left angle bracket upper D right angle bracket . We prove that