NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

80 CASIM ABBAS
Converting from coordinates (s, t) on the half-cylinder to Cartesian coordinates x +
iy = e −(s+it) in the complex plane, we get, with ρ = x 2 + y 2 ,
ds =− 1
(xdx+ ydy) and dt =−1 (xdy− ydx).
ρ2 ρ2 Recall that r(s) = c(s)e κs , where c(s) is a smooth function which converges to a
constant as s →+∞, and we assumed that κ ≤−1/2 and that κ /∈ Z. Another
assumption was that A(r) = O(r). Hence r(s) is close to ρ −κ if s is large (and ρ is
small) and A(r(s)) = O(ρ −κ ). Also recall that γ1(r(s)) = O(1). Summarizing, we
need the expression
A r(s) r(s)ρ −2 ρ = O(ρ −2κ−1 )
to be bounded, which amounts to κ ≤−1/2. The same argument applied to the form
dg ◦ (jτ − jσ ) leads to the same conclusion.
Acknowledgments. I am very grateful to Richard Siefring and Chris Wendl for explaining
some of their work to me. Their results are indispensable for the arguments
in this article. I would also like to thank Samuel Lisi for the numerous discussions we
had about the subject of this article.
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