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Corollary 4. Let A satisfy the hypotheses of Lemma 3. For integers 1≤H≤N , set U=N4/3 , T1=32H2/3 U, T2=T3=32H-1/3 U. The rows ((An)1α1 (An)2α2 (An)3α3 )n∈EN , with α∈3 and αj≤Tj , spanEN . The constant 32 is …

Lemma 3. Let A:4→3 be a surjective linear map, with A= c , where the four coordinates of c are linearly independent over . Let N≥1 and t1,t2,t3 be integers satisfying tj≥3(N-1) (1≤j≤3), ∏j=13(tj-3(N-1)+1)>(4N-3)4. Set EN= …

Lemma 2. For X≥3 , ∑p≤X; χ(p)=1 pp ℓ+δ( X)2ℓ. Consequently, for each integer H≥2 , ∑H<p≤X; p2q; χ(p)=-1 pp ≥ X-Cℓ-Cδ( X)2ℓ-CH, where C is absolute and CH depends only on H .

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Put t_j= T_j . Since H≤N , every T_j≥32N , and T_1T_2T_3=32^3N^4 . In particular t_j≥3(N-1) , and ∏_ j=1 ^3(t_j-3(N-1)+1) >∏_ j=1 ^3(T_j-3(N-1)) ≥(T_1T_2T_3)/(8) =4096N^4>(4N-3)^4. Apply Lemma 3 to the monomial basis of _t .

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