Some results towards the Dittert conjecture on permanents

Let K _n denote the convex set consisting of all real nonnegative n×n matrices whose entries have sum n. For A ∈ K _n with row sums r ₁,⋯,r _n and column sums c ₁, ⋯,c _n, define φ(A)=∏ ⁿ _i=1 ^r _i+∏ ⁿ _j=1c _j-per(A). Dittert's conjecture asserts that the maximum of φ on K _n occurs uniquely at J _n=[1/n] _n×n. In this paper, we prove:(i)if A ∈ K ⁿ is partly decomposable then φ(A) < φ(J _n);(ii)if the zeroes in A ∈ K _n form a block then A is not a φ-maximising matrix;(iii)φ(A) < φ(J _n) unless δ:=per(J _n)-per(A) ≤ (n ⁴ _e ^-2n) and |k-∑i∈αr ⁱ|< √2δk,|k-∑i∈βc _i|<√2δk and ∑i∈α,j∈βaij <k+ √δk for all sets α,β of k integers chosen from {1,2,⋯,n}.