Results on additive decompositions of the primes and their assumptions

The course proves the statements below; each link leads to the statement with its assumptions and its proof. The lessons are self-checked by the writing AI; independent mathematical review is not complete.

• Ostmann's conjecture — Theorem 1.1. If A and B are sets of nonnegative integers with at least two elements each, then the symmetric difference of A + B and the set of primes is infinite. Follows from Lemma 4.1 and Theorem 5.1 of lesson 1.
• A sieve for finitely many shifts — Lemma 3.1. If b_1, ..., b_j are distinct nonnegative integers and c + b_1, ..., c + b_j are all prime for every sufficiently large c in C, then C(Y) << Y/(log Y)^j for large Y, with an implied constant depending only on the shifts.
• Finite summands (Laffer and Mann) — Lemma 4.1. If A and B have at least two elements each and A + B is asymptotically equal to the set of primes, then A and B are infinite.
• Two infinite summands — Theorem 5.1. There are no infinite sets A, B of nonnegative integers such that the symmetric difference of A + B and the set of primes is finite. Proved in lessons 2 to 18.
• Square-root bounds (Elsholtz) — Theorem 3.1. If A and B are infinite and the symmetric difference of A + B and the set of primes is finite, then sqrt(Y)/(log Y)^3 << A(Y), B(Y) << sqrt(Y)(log Y)^2 for Y >= 3.
• Collision stability — Lemma 4.1. For probability measures on the large elements of A and of -B with point masses at most (log X)^b/sqrt(X), the sum over primes p <= sqrt(X)/(log X)^(b+1) of log p times the squared L^2 distances of their reductions from the uniform measures on the residue support S_p and on its complement, plus (1/sigma_p + 1/(1 - sigma_p) - 4)/p, is O(log log X), and every term is nonnegative.
• Decorrelation from translated quadratic characters — Theorem 1.1. As T tends to infinity, the sum over primes with T <= log p <= 2T of (log p)/p times the maximum over t of |E_(x in S_p) chi_p(x - t)| is o(T), where chi_p is the quadratic character modulo p. Proved in lessons 3 to 6.
• Bonami's inequality for signs — Lemma 2.2. For independent uniform random signs and F = sum over S of a_S prod_(i in S) eps_i with real coefficients, (E|F|^(2l))^(1/l) <= sum over S of (2l - 1)^|S| a_S^2 for every integer l >= 1.
• Large quadratic Weyl sums — Lemma 1.1. If a sum of e(alpha j^2 + beta j) over an interval of at most Y/c_1 integers has modulus at least c_1 Y e^(-K), with K large in terms of c_1, Y >= e^(4K) and log(2Y) <= e^K, then ||q alpha|| <= 2e^(8K)/Y^2 for some integer 1 <= q <= 2e^(8K).
• A common rational centre — Proposition 4.2. At least J Z^(-O(1/k)) of the J primes of a biased block have their translations t_p congruent to h/m modulo p, for one pair of coprime integers h, m with 1 <= m <= Z^0.13 and |h| <= X Z^0.15.
• Heath-Brown's quadratic large sieve — Theorem 1.1. For positive integers M, N, complex numbers a_n and eps > 0, the sum over odd squarefree m <= M of |sum over odd squarefree n <= N of a_n (n/m)|^2 is O_eps((MN)^eps (M + N) sum |a_n|^2). Used without proof, as named result N1.
• Split primes for a product of kernels — Proposition 4.1. For the product vw of two small kernels, the primes p <= Q_2 = X^(1/2 - 5 eta) not dividing 2mvw with (vw/p) = 1 satisfy sum (log p)/p >= 0.03 log X; the proof uses the zero-free region with the exceptional zero and Siegel's theorem.
• Decorrelation from characters of higher order — Theorem 1.1. For fixed 0 < alpha < beta, the sum over primes with alpha L <= log log p <= beta L of (1/p) times the maximum over t and over characters lambda of order greater than two of |E_(x in S_p) lambda(x - t)| is o(L). Proved in lessons 7 to 9.
• Kneser's addition theorem — Theorem 3.1. For finite nonempty subsets A, B of an abelian group, |A + B| >= |A + K| + |B + K| - |K|, where K is the stabilizer of A + B (proof of DeVos). Corollary 3.3: a generating set U of a finite abelian group G with 0 in U and |U| >= rho|G| satisfies rU = G for every r >= 2/rho.
• The transfer — Proposition 3.2. Cauchy-Schwarz over the residues of the pivot, extended to positive integers, gives |eta_(j-1)|^2 <= e^(d_j)(eta_j + D_j), where D_j collects the diagonal terms and the off-diagonal part eta_j has the same form as eta_(j-1).
• Uniqueness of frequency histories — Lemma 4.1. For a fixed assignment of the top list and a fixed root frequency there is at most one valid history.
• One-sided character interactions — Lemma 1.2. A factor xi_q(p), with xi_q a nonprincipal character modulo a short prime q and p a long prime, and with no factor in the reverse direction, makes the weighted expectation O(exp(-c e^(alpha L))).
• Mixed flatness of the normalized transforms — Theorem 6.1. Outside a set of primes of harmonic mass o(L) in the band alpha L <= log log p <= beta L, sigma_p = 1/2 + o(1) and the normalized transform g_p of the residue support has correlation o(1) with every product lambda(v) e_p(av) of a multiplicative and an additive character.
• A supply of nonsparse transforms — Theorem 1.1. There is an absolute delta_0 > 0 such that for large L the primes with 0.05 L <= log log p <= 0.9 L, 1/3 <= sigma_p <= 2/3 and gamma_p >= delta_0, where gamma_p is the probability L^1 norm of g_p, satisfy sum 1/p >= 0.15 L. Proved in lessons 10 and 11 from the sieve estimates and coverage.
• A contracting local kernel — Lemma 4.1. For a balanced prime whose transform is sparse (gamma_p < delta_0), the kernel built from the large Fourier coefficients of the support, written in block coordinates between the support and its complement, has norm at most (1 - 1.6/p) times its mean U_p(1,1) = 1 + O(eps_0/p) over the nonzero residues.
• Prime sums twisted by a family of characters — Lemma 2.1. For every fixed D_1 > 0, the sum over primitive characters chi of conductor at most Q_b, without the exceptional character, of |sum Lambda(n) chi(n) phi(n/X) - 1_(chi=1) X Phi(1)| is O(X e^(-D_1 L)); the proof uses the explicit formula, the zero-free region and the log-free zero-density estimate.
• The tree comparison over a finite field — Lemma 1.2. A single binary-tree diagram of transforms over F_q with independent uniform leaf variables has E|W|^2 <= 3^r. For two diagrams of depth l >= 2 whose r leaves are partitioned into b <= 3r/4 paired blocks, with leaf variables uniform subject to equal products over paired blocks, |E W_1 conj(W_2)| <= C_l(eps_q + q^(-1/4)). Proved in lessons 12 and 13.
• A uniform Mellin estimate — Proposition 1.1. For every multiplicative character eta and every a in F_q, the sum over all multiplicative characters chi of the squared Fourier coefficient at a of P_chi chi eta, where P_chi are the Mellin coefficients of a bottom-pair autocorrelation, is at most q/(q - 1)(eps_q^4 + sqrt(3) q^(-1/2)); each nonidentity affine map of F_q arises in at most two ways.
• The exact transfer — Proposition 5.1. For 1 <= j <= k, e^(-c_g)|eta_(j-1)|^2 <= D_j + eta_j, with a nonnegative diagonal D_j and a real eta_j.
• The comparison estimates — Proposition 6.1. For one assignment, the weighted mean square of the depth-l amplitude is at most exp(r(Delta_0 + Cm) + o_k(m)); for two assignments with l >= 2 whose overlap graph has at most 3r/4 components, the weighted correlation is at most exp(-omega_k(m)) with omega_k(m)/m tending to infinity. Proved in lessons 15 to 17.
• Primes in progressions in short logarithmic intervals — Lemma 2.1. For log M_* <= e^(mu L), a residue a coprime to M_* and an interval I of length at most 1 inside [e^(a_0 L), e^(a_1 L)], the sum of 1/p over primes p = a mod M_* with log p in I equals 1/phi(M_*) times the integral over I of (1 - chi_*(a) e^((beta_* - 1)t)) dt/t, up to O(exp(-c e^(delta' L))), where chi_* is the possible exceptional real character and beta_* its real zero.
• The signed comparison — Proposition 2.2. Under the hypotheses of the pair estimate of lesson 14, the weighted correlation of two assignments is at most eps_k(L)^m exp(O_k(m)) + O(exp(-e^(0.00125 L))) = exp(-omega_k(m)), from the tree comparison at every spectator prime.
• The norm comparison — Proposition 3.3. Under the hypotheses of the norm estimate of lesson 14, the weighted mean square of the depth-l amplitude is at most exp(r(Delta_0 + Cm) + o_k(m)), with C independent of large k and of the gap constants.
• The lower bound — Proposition 2.2. For large L, -log|eta_j| <= 2^j B_* m + (2^j - 1)(c_g + log 2) < B 2^j m for 0 <= j <= k.
• The norm of the common regular transform — Proposition 3.1. For large L, the expectation of the sum over 0 < |s| <= V_k of |G_R(s/d)|^2 is O(V_k) with an absolute implied constant.

Reading order

Eighteen lessons, to be read in order. The first two set up the problem, the residue supports and collision stability. Lessons 3 to 6 prove that translated quadratic characters do not bias the supports, and lessons 7 to 9 do the same for characters of higher order. Lessons 10 and 11 use coverage of every large prime to show that many supports have nonsparse Fourier transforms. Lessons 12 and 13 prove a comparison estimate for binary trees over finite fields, and lessons 14 to 18 build a positive statistic from the nonsparse transforms and prove that it is both large and small. The analytic inputs are proved in the programme courses on the Riemann zeta function and on Dirichlet L-functions; Heath-Brown's quadratic large sieve is used without proof as named result N1.