Sum-difference exponents for boundedly many slopes, and rational complexity

1.3. Notation.

• The exposition uses Shannon-entropy formulations, conditional entropy, and mutual information, exploiting entropy inequalities and entropic Plünnecke–Ruzsa calculus.The sum-difference constant also has an equivalent finite-set formulation, but the entropy formulation is used for the paper’s arguments.Source: §1.1 · pp. 1–2, §1.3 · p. 6

1.1. The arithmetic Kakeya conjecture.

• The arithmetic Kakeya conjecture asserts that, as the number of slopes R grows, the constants SD(R; s) converge to 1; establishing it would imply full Minkowski and Hausdorff dimensions in every dimension, currently known only for d ≤ 3.Source: §1.1 · p. 2, §1.2 · p. 3

1.2. Asymptotic behavior.

• For a fixed finite slope set R, the paper instead studies variation in the slopes and predicts convergence of SD(R; s) to 2 as expressing s rationally from R becomes more complex.The analysis is normalized to slope sets containing 0, 1, and ∞, using projective invariance.Source: §1.1 · p. 2, §1.2 · p. 3
• Rational complexity D(R; s) is the least degree-and-coefficient scale of a rational expression representing s from the slopes in R.Its definition uses integer-coefficient polynomials P and Q of degree at most D and coefficient magnitude at most 2^D, with Q nonzero; for R={0,1,∞}, it corresponds to the size of a reduced numerator and denominator.Source: §1.2 · p. 3
• Theorem 1.3 gives 2 − c2/D(R; s) ≤ SD(R; s) ≤ 2 − c1/D(R; s), with absolute constants c2 > c1 > 0.Thus, for boundedly many slopes, the exponent approaches 2 at a rate controlled by rational complexity.Source: §1.2 · p. 4
• For three slopes, the paper also gives logarithmic asymptotics suggested by AlphaEvolve experiments and develops a continuous-limit lower bound as a/b approaches a real α with b→∞.The continuous-limit expression involves a variational constant cα, and the authors conjecture that the corresponding upper bound is asymptotically sharp.Source: §1.2 · pp. 4–5

• The entropic Ruzsa distance is non-negative, symmetric, satisfies an entropy-difference bound, and obeys Ruzsa-type triangle inequalities under stated independence conditions.It is not a true metric because d[X; X] can be non-zero.Source: §2 · p. 7
• The auxiliary quantity g(a) = H[X−aX′]−H[X] measures entropy growth under a rational sum-difference with an independent copy X′.Its basic identities and inequalities control signs, reciprocals, products, and sums of rational coefficients.Source: §2 · p. 7
• For slopes represented by bounded-degree, bounded-coefficient polynomials, g is controlled by the maximum g(r) over the generating slopes, with dependence on rational complexity and the number of slopes.The proof proceeds from monomial estimates to polynomial expressions and then to ratios of polynomials.Source: §2 · pp. 7–8
• The review records an entropic Balog–Szemerédi–Gowers lemma for conditionally independent rational-valued variables, obtained by averaging over the conditioning variable.The stated inequalities are cited from prior lemmas and applied in the unconditional case after averaging.Source: §2 · p. 8

3. A REDUCTION TO THE INDEPENDENT CASE

• Theorem 3.2 relates the independent sum-difference constant f(R; s) to SD(R; s), enabling the analysis to work with independent random variables.The proof uses entropic Balog–Szemerédi–Gowers arguments to replace coupled variables while preserving the required controls.Source: §3 · p. 9, §3.1 · p. 9

3.1. First step: obtaining a good configuration.

• A good configuration initially exists but need not have an invertible projection or controlled X- and Y-doubling.The subsequent steps add these properties without abandoning the configuration framework.Source: §3.1 · pp. 9–10

3.2. Second step: ensuring an invertible projection.

• Proposition 3.6 constructs a good configuration with an invertible projection by randomly selecting one fiber representative for each value of the output projection.The resulting shuffled variables retain the relevant distributional properties and yield an invertible projection.Source: §3.2 · p. 11

3.3. Third step: controlling one doubling constant.

• Proposition 3.7 adds an auxiliary variable to obtain controlled X-doubling while preserving goodness and invertibility.The entropic Balog–Szemerédi–Gowers lemma supplies the core doubling-control mechanism.Source: §3.3 · p. 12

3.4. Fourth step: controlling a second doubling constant.

• Proposition 3.8 and Lemma 3.9 together preserve controlled X-doubling while adding controlled Y-doubling, yielding simultaneous control.This resolves the technical concern that the second reduction might destroy the first doubling property.Source: §3.4 · pp. 12–13

3.5. Fifth step: controlling projections of conditionally independent variables.

• The final estimates bound projections involving conditionally independent copies for every relevant slope r, completing the ingredients for the independent-case reduction.Proposition 3.11 packages these projection bounds for configurations supplied by Corollary 3.10.Source: §3.5 · pp. 13–15

4. PROOF OF THEOREM 1.3

• For s = a/b, the lower-bound construction uses independent uniform variables on an interval whose scale is set by |a| and b.The projection with slope a/b is injective, producing entropy of order log(2 + |a| + |b|).Source: §4 · p. 15
• If the target projection were non-injective, polynomial combinations of slopes in R would express s with lower complexity, contradicting its assumed rational complexity.This injectivity argument supplies the matching lower bound after the entropy estimates are inserted into the main inequality.Source: §4 · p. 16
• The upper bound reduces to controlling the independent constant f(R; s) when s has complexity D.The proof introduces polynomial-combination sets and propagates bounds through the function g(a) using Proposition 2.1(iv) and entropic Ruzsa inequalities.Source: §4 · p. 16
• The argument starts from entropy bounds at slopes in R and transfers them to s through rational-combination control, yielding the claimed upper estimate.The resulting bound depends on the complexity through the exponent appearing in the estimate for g(s).Source: §4 · pp. 15–16

5. A CONTINUOUS LIMIT

• Theorem 1.4’s lower bound is reduced to evaluating entropies of lattice-supported variables generated from a smooth compactly supported function.The proof allows implied constants to depend on the parameters specified in the theorem.Source: §5 · p. 17
• Comparing the resulting lower bound with Theorem 1.3 shows that c_α is strictly positive and finite and yields the claimed expression (1.8).The proof approaches the infimum defining c_α using arbitrary test functions.Source: §5 · p. 17
• Poisson summation, Bezout’s theorem, dominated convergence, and Riemann integrability control lattice representations and approximation errors.The argument uses the rapid Fourier decay of the smooth density and an error term of order O(1/b) before passing to the limit.Source: §5 · pp. 17–18
• The projection π1(X, Y) = X + Y has entropy log b + h(f1) + o(1), which directly yields equation (5.1).This computation is one of the limiting entropy identities used in the theorem.Source: §5 · p. 18