On the exceptional set in Goldbach’s problem in short intervals

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On sums of squares of primes (2006, methodology): “The sieve method We now show how a function ρ having properties (i)–(v) above can be constructed using the sieve method originating in [6] and developed in [1, 7].”
Graphs and matrices of maximal energy (2006, background): “, [1], Theorem 3), for n su¢ ciently large, there exists a prime p such that p 1 (mod 4) and p n+n: Suppose n is large and x some prime p n+n=2: The average number of edges induced by a set of size n in Gp is”
On Weyl sums over primes and almost primes (2006, unknown): “5 playing the roles of Lemmas 4 and 9 in [2].”