Problem 32 — Visual explainer (undergrad)
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Problem statement
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Is there a set $A\subset\mathbb{N}$ such that\[\lvert A\cap\{1,\ldots,N\}\rvert = o((\log N)^2)\]and such that every large integer can be written as $p+a$ for some prime $p$ and $a\in A$?

Can the bound $O(\log N)$ be achieved? Must such an $A$ satisfy\[\liminf \frac{\lvert A\cap\{1,\ldots,N\}\rvert}{\log N}> 1?\]

Picture
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Target integer N as "prime + offset":

N  =  p  +  a
     ^     ^
   prime  pick a from A

Think: A is a small "menu of offsets" that helps you hit all large N.

Question: what to look at
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- Restate the problem as: given the constraint, what asymptotic bound or
  structure must follow?
- Use the comments as benchmarks (best known bounds / constructions).

Context
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Area hint: additive number theory (primes) / sieve methods.

Notation (if it appears above)
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- `o(g(n))` means "grows strictly smaller than g(n)" (ratio → 0 as n→∞).
- `O(g(n))` means "at most a constant times g(n)" for large n.

Benchmarks / known results (from comments)
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- Erd\H{o}s proved that such a set $A$ exists with $\lvert
  A\cap\{1,\ldots,N\}\rvert\ll (\log N)^2$ (improving a previous result of
  Lorentz who achieved $\ll (\log N)^3$).
- Wolke has shown that such a bound is almost true, in that we can achieve
  $\ll (\log N)^{1+o(1)}$ if we only ask for almost all integers to be
  representable.
- Kolountzakis improved this to $\ll (\log N)(\log\log N)$, and Ruzsa further
  improved this to $\ll \omega(N)\log N$ for any $\omega\to \infty$.
- The answer to the third question is yes: Ruzsa has shown that we must
  have\[\liminf \frac{\lvert A\cap\{1,\ldots,N\}\rvert}{\log N}\geq
  e^\gamma\approx 1.781.\]This is discussed in problem E1 of Guy's collection,
  where it is stated that Erd\H{o}s offered \$50 for determining whether
  $O(\log N)$ can be achieved.