Petr M. Akhmet'ev's work on the Arf-Kervaire invariant problem, June, 2009.

Akhmet'ev has some preprints leading to a theorem that θj does not exist for large j. The key papers are in Russian with shorter summaries in English. His approach is quite different from ours, and uses a geometric method suggested by some results of Eccles.

A geometric solution to the Kervaire Invariant One problem,
slides for a talk given by him at Princeton on May 19, 2009.
My understanding of Akhmet'ev's program,
a two page synopsis.
Akhmet'ev's Compression Theorem (6 pages, July 8, 2009)
This result is stated without proof on page 19 of Akhmet'ev's slides and at the end of my overview document. It is referred to there as the "desuspension theorem", and in some versions of the paper as the "retraction theorem". It is critical to determining which θjs can be excluded by his method. The proof given in his 2009 paper (in Russian) is over 20 pages.
This document gives a complete proof and some explicit computations of some of the numbers referred to in the theorem.
K-theory of S7/Q8 and a counterexample to a result of P.M. Akhmet'ev
(January 18, 2010, 6 pages) by Peter S.Landweber.
This short paper gives a counterexample to Proposition 41 (37 in the English translation) in Akhmet'ev's Hopf invariant paper, which concerns the orbit space S7/Q8 of the action of the quaternion group Q8 on the 7-sphere. The assertion is that any map to it from a smooth closed 7-manifold with a certain condition on its normal bundle must have even degree. Landweber uses classical K-theory to show that the identity map satisfies the normal bundle condition and is therefore a counterexample to the assertion of even degree.
Akhmet'ev's arXiv preprints, some of which are listed below.
Geometric approach towards stable homotopy groups of spheres. I. The Hopf Invariant (English, 48 pages, August, 2011), P. M. Akhmetiev.
The purpose of this paper is to reprove a weaker version Adams' Hopf invariant one theorem using geometric methods similar to those used to deal with the Arf-Kervaire invariant.
Geometric approach towards stable homotopy groups of spheres II. Kervaire Invariant, Janaury, 2012 version (English, 104 pages), P. M. Akhmetiev. It was the subject of the lecture he gave at the Edinburgh conference of April, 2011. This is his main paper.
"We prove that for sufficiently large n, n = 2j - 2, an arbitrary skew-framed immersion in Euclidean n-space Rn has zero Kervaire invariant. Additionally, for j >= 12 (i.e., for n >= 4094) an arbitrary skew-framed immersion in Euclidean n -space Rn has zero Kervaire invariant if this skew-framed immersion admits a compression of order 16."
My commentary on Landweber's partial translation, July 8, 2009.
Earlier published papers
Codimension one immersions and the Kervaire invariant one problem, Peter J. Eccles, 1981. This paper provides the basis of Akhmet'ev's program.
A geometrical proof of Browder's result on the vanishing of the Kervaire invariant, (English, 6 pages, 1998)  Pyotr M. Akhmetiev and Peter J. Eccles. Abstract: Browder proved that the Kervaire invariant of a framed manifold of dimension n=4k+2 vanishes if n+2 is not a power of 2. We give a geometric proof using a characterization of it in terms of multiple points of immersions.
The relationship between framed bordism and skew-framed bordism, (English, 9 pages, 2005), Pyotr M. Akhmetiev and Peter J. Eccles
Geometric approach towards stable homotopy groups of spheres. Kervaire Invariants. II (English translation of a paper published in 2007, 16 pages), P. M. Akhmetiev. My commentary on this paper.