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Source. Conjectures 13, 14 and 15, display (5.1) and the paragraphs around them, p. 11, of Javier Cilleruelo and Andrew Granville, Lattice points on circles, squares in arithmetic progressions and sumsets of squares, Additive Combinatorics, CRM Proceedings and Lecture Notes 43 (Amer. Math. Soc., 2007), 241-262. Labels and pages are those of the arXiv preprint math/0608109v1 identified on the source card.

Statement

Conjecture 13 (p. 11). For any $\alpha<1/2$ there is a constant $C_\alpha$ such that for any $N$,

$$\#\{(a,b):\ a^2+b^2=n,\ N\le|b|<N+n^\alpha\}\le C_\alpha .$$

The print leaves $n$ unquantified; this page reads the bound as holding for every $n$. The special case $N=0$ is display (5.1): $\#\{(a,b): a^2+b^2=n,\ |b|<n^\alpha\}\le C_\alpha$.

Conjecture 14 (p. 11). The number of lattice points $\{(x,y)\in\mathbb Z^2: x^2+y^2=R^2\}$ in an arc of length $R^{1-\epsilon}$ is bounded uniformly in $R$.

Conjecture 15 (p. 11). The same, for an arc of length $R^{1-\epsilon}$ around the diagonal.

Equivalence (p. 11). The paper states that Conjecture 13 and (5.1) are equivalent to Conjectures 14 and 15 respectively. It argues that Conjectures 13 and 14 rephrase one another and imply (5.1) and Conjecture 15; conversely, points of $x^2+y^2=R^2$ on an arc of length $R^{1-\epsilon}$ are rotated, by multiplying by the conjugate of one of them, to points with $|b_j|\ll R^{1-\epsilon}$, contradicting (5.1), and multiplying further by $1+i$ gives points of $x^2+y^2=2R^2$ on an arc around the diagonal, contradicting Conjecture 15. All four statements are thus presented as equivalent. None is proved in the paper.

Known ranges (p. 11). The paper states that (5.1) is simple to prove for any $\alpha\le1/4$, and Conjecture 13 for $\alpha\le1/4$ with $N\ll n^{1/2-\alpha}$, but that the authors cannot prove (5.1) for any $\alpha>1/4$. The unconditional result on short arcs is Theorem 13.

The flowchart on p. 15 labels the plain-arc box 15 and the diagonal box 14, the reverse of the text on p. 11; this page follows the text. Theorem 16 (p. 14) states that Conjecture 13 implies Conjecture 19, on $L^4$ norms of trigonometric polynomials with frequencies in $\{N^2,\ldots,(N+N^\alpha)^2\}$; the flowchart draws this as an arrow from its diagonal box to Conjecture 19.

Proof pointer

The equivalence argument is the paragraph after Conjecture 15, p. 11; the conjectures themselves are not proved.

Dependencies

None. Read depth: claims checked on p. 11; the flowchart on p. 15 was compared with the text.

Bears on

No Erdős problem in the corpus.